Van der Waerden rings

Accepted Manuscript Van der Waerden Rings

Dieter Remus, Mihail Ursul

PII: DOI: Reference:

S0166-8641(17)30353-X http://dx.doi.org/10.1016/j.topol.2017.07.010 TOPOL 6181

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Topology and its Applications

Received date: Revised date: Accepted date:

21 April 2016 8 June 2017 19 July 2017

Please cite this article in press as: D. Remus, M. Ursul, Van der Waerden Rings, Topol. Appl. (2017), http://dx.doi.org/10.1016/j.topol.2017.07.010

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VAN DER WAERDEN RINGS DIETER REMUS AND MIHAIL URSUL Respectfully Dedicated to the Memory of Our Friend W. Wistar Comfort Abstract. The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7]. A compact ring (R, T ) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R. In this paper the study of this class of compact rings is continued. We prove that  an Abelian compact van der Waerden ring R has the form R = ( i∈I Ri ) × L, where L is a compact radical ring with radical topology and Ri (i ∈ I) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings.

1. Introduction The relation between algebraic and topological properties of a topological algebraic system is a central topic in topological algebra. A classical result on this topic is the theorem of Cartan and van der Waerden stating that if G is a compact connected semisimple Lie group, H a Lie group and ϕ : G → H a homomorphism whose image is bounded, then ϕ is continuous ([5], [35]). The following notion was introduced in [7](p. 27) : A compact ring (R, T ) is a van der Waerden ring if every homomorphism ϕ : (R, T ) → (K, U) with (K, U ) a compact ring is continuous. It was noted in [7](p. 27) that this notion was suggested by van der Waerden’s paper [35]. In particular, a classification of semisimple van der Waerden rings was obtained in [7](Theorem 4.13). Date: July 19, 2017. 1991 Mathematics Subject Classification. Primary 22H11, Secondary 54A25. Key words and phrases. semisimple ring; local ring; formal power series ring; topological p-ring; van der Waerden ring; radical topology; p-adic integers; topologically finitely generated compact ring; indecomposable ring; variety of rings; linear ring; totally bounded topological ring; Kaplansky representation. Corresponding author : Dieter Remus. 1

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We note that the class of compact rings with a unique compact ring topology – see [37], [3] – is close to the class of van der Waerden rings. Indeed, a compact ring (R, T ) is a van der Waerden ring (compact ring with a unique compact ring topology) provided each totally bounded (compact) ring topology T  on R coincides with T . This implies that every van der Waerden ring is a ring with a unique compact ring topology. The compact ring Fω2 has a unique compact ring topology, but it is not a van der Waerden ring. Every van der Waerden ring is totally disconnected (Theorem 3.4) and metrizable (Corollary 4.5). The class of van der Waerden rings is closed under extensions (Theorem 3.20) and supports some operations: (1) for each van der Waerden ring (R, T ) and each n ∈ N, the ring M (n, R) is a van der Waerden ring with respect to the canonical topology of M (n, R) (Theorem 3.25). (The case, when (R, T ) is a ring with identity, was treated in [7](Theorem 6.7).) (2) It is proved that if (R, T ) is a compact ring with identity then the formal power series ring (R[[X]], T0 ) with the natural compact topology T0 is a van der Waerden ring if and only if (R, T ) is a van der Waerden ring (Theorem 5.20). Recall [32](p. 132) that a compact ring R is said to have radical topology provided the family {J(R)n |n ∈ N} is a fundamental system of neighborhoods of zero of R. Theorem 5.5 gives a complete characterization of van der Waerden rings with open radical: A compact ring R with open radical is a van der Waerden ring if and only if R is a ring with radical topology. Every van der Waerden ring R has a lot of van der Waerden subrings, since every retract and every open subring of a van der Waerden ring is a van der Waerden ring (Corollary 3.2 and Theorem 3.23). Every van der Waerden ring R can be embedded into a van der Waerden ring with identity; if the characteristic of R is finite, then R can be embedded into a van der Waerden ring of the same characteristic (Theorem 5.14). Partially we answer Question 4.10 posed in [6]: What is the structure of van der Waerden rings R (in particular if R is a commutative ring with identity)? We give a complete characterization of Abelian van der Waerden rings (Theorem 5.16). This class contains the class of compact rings without nonzero nilpotent elements and the class of commutative compact rings. It follows from [32](Chapter 2, Theorem 6.34) that every compact topologically finitely generated ring with identity is a van der Waerden ring. We study in this context the question how to distinguish compact finitely generated rings from van der Waerden rings in the case

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of semisimple rings. Theorem 5.31 furnishes a complete answer to this question. We construct an example of a semisimple van der Waerden ring without a finite number of topological generators (Example 5.33) and two examples of indecomposable van der Waerden rings which are not topologically finitely generated (Examples 5.34 and 5.35). Theorems 5.5 and 5.12 emphasize relations between van der Waerden rings and compact rings with open radical. The following classes of compact rings are studied in section 6: Rings of type(F), rings of type(AF) and strongly complete rings. They are closely related to van der Waerden rings. Among other things it is proved that every cofinite ideal of a compact ring R is open if and only if R is of type(AF) (Theorem 6.4). This furnishes a characterization of van der Waerden rings with identity (of finite characteristic) (Corollary 6.6). In section 7 conditions for compact rings R are given such that R is van der Waerden ring. Besides this, the study of van der Waerden rings with identity is reduced to the study of indecomposable van der Waerden rings. 2. Notations, Conventions, and Auxiliary Results If A is a subset of B, we write A ⊂ B. ω stands for the set of all natural numbers, N stands for the set of all numbers 1, 2, 3, . . ., P stands for the set of all prime numbers, Z stands for the group of integers, R stands for the group of real numbers, T stands for the group R/Z. [m, n], where m, n ∈ ω, m ≤ n stands for the set {m, m + 1, . . . , n}, Zp , where p ∈ P, stands for the group of p-adic numbers, Z(n) stands for the cyclic group consisting of n elements. If H is a subgroup of a group G, then [G : H] stands for the index of H in G. A subgroup H of a group G is called cofinite if [G : H] < ∞. A ring R means associative not necessarily with identity. Its center is denoted by Z(R). M (n, R), where n ∈ N and R a ring, stands for the ring of n × n matrices with coefficients from R. If R is a ring and n ∈ N, then [n]R = {a1 + · · · + an |ai ∈ R, i = 1, . . . , n} and R[n] = {a1 · · · an |ai ∈ R, i = 1, . . . , n}. Rn means the subset of R consisting of all finite sums of elements from R[n] , and we set R0 = {0}. Surely Rn is an ideal of R. For A, B ⊂ R, let A·B = {ab|a ∈ A, b ∈ B}. AB means the subset of R consisting of all finite sums of elements from A · B. The subring of a ring R generated by a subset S is denoted by S, and the subgroup of the additive group generated by S is denoted by S+ . We will say that a ring R has a finite characteristic provided

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there exists k ∈ N such that ∀x∈R [kx = 0]. The least number with this property is called the characteristic of R and is denoted charR. Fpn where p ∈ P, n ∈ N stands for the Galois field consisting of pn elements. A ring is R called Abelian if every idempotent of R is central ([4], 1.1.12). Let R be a ring, and let e ∈ R be an idempotent. Then (1 − e)R := {a − ea|a ∈ R} resp. R(1 − e) := {a − ae|a ∈ R}. The additive group (R, +) has the Peirce decompositions R = eR ⊕ (1 − e)R and R = Re ⊕ R(1 − e) (for further details see [16], chapter III, §7 and [21], §21). An ideal I of a ring R is called quasiregular provided for each x ∈ I there exists y ∈ I such that x + y + xy = 0 = x + y + yx. The Jacobson radical of a ring R is denoted by J(R) or, briefly, by J. (The Jacobson radical of a ring has many characterizations. One of them is: the Jacobson radical of a ring is the largest quasiregular ideal of R.) By a radical ring we mean a ring being radical in the sense of Jacobson. Since a ring R is radical iff each its element is quasiregular, we call sometimes a radical ring a quasiregular ring. For a compact ring R the following holds: R is radical ⇔ R is quasiregular ⇔ R is topologically nilpotent [32](Chapter 2, Theorem 6.6). A ring R is called semisimple if J(R) = {0}. A ring R with identity is said to be a local ring if R has a unique maximal left ideal, or, equivalently, if R has a unique maximal right ideal. A nonzero ring is local if and only if R/J(R) is a division ring (for further details see [21], §19). By R0 we denote the connected component of zero of a topological ring R, i.e., the maximal connected subset containing zero. If (R, T ) is a topological ring and I an ideal of R, then T I stands for the induced topology on I and T /I for the quotient topology on R/I. The closure of a subset A of a topological space is denoted by A. R ∼ =top S means that the topological rings R and S are algebraically and topologically isomorphic. A topological ring is called linear if it has a fundamental system of neighborhoods of zero consisting of ideals. Some authors call linear rings linearly topologized. A Hausdorff topological ring is called totally bounded if it is Haudorff and precompact. For the definition of the ring of all formal power series over a ring, see, e.g., [21](p. 9) and [11]. A natural compact topology on the ring of all formal power series R[[X]]with coefficients from a compact totally disconnected ring (R, T ): Let (R, T ) be a compact totally disconnected ring. Consider the family B of ideals of R[[X]] of the form V = V + V X + V X 2 + · · · + V X n + RX n+1 + · · · , where V is an open ideal of R. It is clear that V is a cofinite subgroup of R[[X]]. We have to show that V is an ideal of R[[X]]. Let f = a0 + a1 X + · · · + an X n + an+1 X n+1 + · · · ∈ V and

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g = b0 +b1 X +· · ·+bn X n +bn+1 X n+1 +· · · ∈ R[[X]]. Then by definition k f g = Σ∞ k=0 dk X , where dk = Σi+j=k ai bj , k ∈ ω. If k ≤ n and i + j = k then i ≤ n, hence ai bj ∈ V and so dk ∈ V. This means that f g ∈ V ; analogously, gf ∈ V . Clearly, the family B is a filter base on R[[X]] with ∩B = {0}. Therefore B gives a Hausdorff ring topology T0 on R[[X]]. In order to show that (R[[X]], T0 ) is a compact ring it suffices to note that the group isomorphism f : Rω → R[[X]], (a0 a1 . . .) → a0 + a1 X + · · · , is topological. Motivated by [1] a compact ring R is called of type (AF) resp. type (F) if for every n ∈ N, R has at most a finite number of ideals of index n resp. R has at most a finite number of open ideals of index n. A profinite ring means a totally disconnected compact ring, equivalently, an inverse limit of a system of finite rings. By analogy to [28](p. 120) a profinite ring (R, T ) is called strongly complete if every cofinite ideal of R is open. (This means: The topology T is the finest linear totally bounded ring topology on R.) For van der Waerden ring we write vdW-ring shortly in the following. Lemma 2.1. Let G be a nontrivial compact, connected Abelian group. Then there exists a closed subgroup H of G such that G/H ∼ =top Tn for some n ∈ N. Proof. Apply [14](24.25) and Pontryagin-van Kampen duality.



Lemma 2.2. ([14](25.31)) Let G be compact group, α a nonzero cardinal, and suppose that G contains a subgroup H which is topologically isomorphic with Tα . Then H is closed, and G is topologically isomorphic with H × G/H. Theorem 2.3. ([17], Theorem 8; [36](32.3)) In a compact ring R the connected component R0 of 0 satisfies R0 · R = R · R0 = {0}. Theorem 2.4. ([17], Theorem 16; [36](32.6),(32.7)) A nonzero, compact, semisimple ring R  has an identity element and is topologically isomorphic to a product i∈I Ri with each Ri a finite simple ring; if R is commutative each ring Ri is a finite field, otherwise each is a full matrix ring over a finite field. Corollary 2.5. Every closed ideal I of a semisimple compact ring R has the form Re, where e is a central idempotent of R. Proof. Since R is semisimple, the ideal I has the same property. By Theorem 2.4 it has an identity e. Obviously, I = Re and e is a central idempotent of R. 

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The following result is a slight improvement of Lewin [23](Lemma 1). Lemma 2.6. Let R be a ring, and let S be a cofinite subring with 2 [R : S] = n. Then S contains an ideal J of R of with [R : J] ≤ nn +1 . Proof. Consider firstly that S does not contain nonzero ideals. We will 2 show that |S| ≤ nn . Indeed, let R = S ∪ (r1 + S) ∪ · · · ∪ (rn−1 + S). Denote K = {1, r1 , . . . , rn−1 }, where 1 the identity if the ring has an identity and the formal identity if R is a ring without identity. Consider for each s ∈ S the mapping ψs : K × K → R/S, (ri , rj ) → ri srj + S. 2 Assume that |S| > nn = |(R/S)K×K |. Then there exist s1 , s2 ∈ S, s1 = s2 such that ψs1 = ψs2 , hence ri (s1 − s2 )rj ∈ S for all i, j ∈ [0, n − 1]. It follows easily that 0 = (s1 − s2 ) ⊂ S, where (s1 − s2 ) is the 2 ideal generated by s1 − s2 , a contradiction. It implies that |R| ≤ nn +1 . Let R be arbitrary, and let S be a subring of index n. Let J be the sum of all ideals contained in S. Then S/J has index n in R/J, hence 2 2 |S/J| ≤ nn . Thus |R/J| ≤ nn +1 .  Lemma 2.6 can be used (instead of [23](Lemma 1)) in the proof of the following theorem. Theorem 2.7. ([23], Theorem 2) A subring of finite index in a finitely generated ring is again finitely generated. In what follows we fix a faithful indexing {Rn |n < ω} of the set of all matrix rings over finite fields (see [7](4.12)). We use Theorem 2.4 of Kaplansky to write each compact semisimple ring R in the form  R = n∈I Rnαn for suitable I ⊆ ω and cardinal numbers αn . This representation is called Kaplansky representation. Theorem 2.8. ([7], Theorem 4.13) LetR be a compact semisimple ring with Kaplansky representation R = n∈I Rnαn . In order R to be a vdW-ring it is necessary and sufficient that each αn be finite. Theorem 2.9. ([13], Theorem 17.2) A bounded Abelian group is a direct sum of cyclic groups. From [7](Lemma 4.3) we get as a special case Lemma 2.10. Let R be a compact ring. If R is a vdW-ring, then every ideal of finite index is open. If R has an identity, then the converse is true. In particular, every vdW-ring is strongly complete.

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Theorem 2.11. ([32], Chapter 2, Theorem 4.20) If R is a right topological ring, I a compact ideal in R and x + I is an idempotent of the quotient ring R/I, then there exists an idempotent e ∈ x + I. Our primary reference for terminology and notation is the book [32]. 3. First Properties of the Class of van der Waerden Rings The following result is an easy consequence of the definition of a vdW-ring. Lemma 3.1. A Hausdorff continuous homomorphic image of a vdWring is a vdW-ring. Recall that a subring S of a ring R is called a retract if there exists a homomorphism f : R → S such that f (s) = s for all s ∈ S. Corollary 3.2. Every retract S of a vdW-ring R is a vdW-ring. Proof. Let p : R → S be the projection of R on S. Since S ⊂ R, p is continuous by the definition of a vdW-ring. Now apply Lemma 3.1.  Lemma 3.3. If (R, T ) is a compact ring and its connected component R0 of 0 is T, then it is not a vdW-ring. Proof. By Lemma 2.2 (R, +) = S ⊕ R0 is a topological direct sum of a totally disconnected subgroup S and R0 . Let {Uα }α∈Ω be a fundamental system of neighborhoods of zero of (S, T S ) consisting of subgroups, and let {Wγ }γ∈Γ be a fundamental system of neighborhoods of zero of (R0 , T R0 ). We can assume without loss of generality that each Wγ does not contain any nonzero subgroup. We claim that for any α ∈ Ω there exists β ∈ Ω such that R·Uβ ⊂ Uα and Uβ · R ⊂ Uα . Indeed, fix any γ ∈ Γ. There exist β ∈ Ω and δ ∈ Γ such that R · (Uβ + Wδ ) ⊂ Uα + Wγ and (Uβ + Wδ ) · R ⊂ Uα + Wγ . If p is the projection of R onto R0 , then p(R · Uβ ) = {0}. Hence R · Uβ ⊂ Uα . Similarly, Uβ · R ⊂ Uα . Let {Hλ }λ∈Λ be a fundamental sytem of neighbooords of zero of (R0 , V) for a compact group topology on R0 , where T R0 = V. (By [15] T has c many compact group topologies which are pairwise nonhomeomorphic.) Obviously, the family {Uα +Hλ |(α, λ) ∈ Ω×Λ} defines a compact ring topology T1 on R and T1 = T . Hence (R, T ) is not a vdW-ring.  Theorem 3.4. Any vdW-ring R is totally disconnected.

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Proof. Assume on the contrary that R0 = {0}. By Lemma 2.1 there exists a closed subgroup H of R0 such that the quotient group R0 /H is topologically isomorphic to T. Since by Theorem 2.3, R · H = H · R = {0}, H is an ideal of R. By Lemma 3.1, R/H is a vdW-ring. A contradiction with Lemma 3.3.  Lemma 3.5. Let R be a compact ring such that R0 is a product of a finite number of copies of T. Then there exists a closed ideal J = R such that R = J + R0 . Proof. By Lemma 2.2 there exists a compact totally disconnected subgroup K of R such that R = K ⊕ R0 , a topological direct sum of K and R0 . We will find an ideal I ⊂ K of R open in K. Let W be a neighborhood of zero of R0 which does not contain any nonzero subgroup. Let U be an open subgroup of K such that U · R ⊂ K + W , R · U ⊂ K + W , and R · U · R ⊂ K + W . We claim that U · R ⊂ K. Indeed, let u ∈ U, r ∈ R. Then there exist k1 ∈ K and w1 ∈ W such that ur = k1 + w1 . If n ∈ Z, then there exist k2 ∈ K, w2 ∈ W such that n(ur) = k2 + w2 = nk1 + nw1 . Thus nw1 ∈ W , hence w1 = 0. Similarly, R · U ⊂ K and R · U · R ⊂ K. If I = (U ) be the ideal of R generated by U , then I is open in K. Let n = [K : I] and let J = {x ∈ R|nx ∈ I}. Then K ⊂ J. The mapping fn : R → R, x → nx is continuous and J = fn−1 (I), hence J is closed. It is easy to check that J is an ideal of R and R = J + R0 . The additive group of the factor ring J/I is torsion, hence totally disconnected. Therefore, J is totally disconnected. Hence J = R.  Lemma 3.6. Let R be a compact ring and R0 = {0}. Then there exists a closed ideal I = R such that R = I + R0 . Proof. By Lemma 2.1 there exists a closed subgroup H of R0 such that R0 /H ∼ =top Tn for some n ∈ N. Since R · H = H · R = {0}, H is an ideal of R. The component of zero of R/H is R0 /H. By Lemma 3.5 there exists a proper closed subring L of R/H such that R/H = L + (R0 /H). Since L is totally disconnected, L = R/H. If ϕ : R → R/H is the canonical homomorphism, then R = ϕ−1 (L) + R0 . Since ϕ−1 (L) is proper, this finishes the proof.  Corollary 3.7. If R is a compact ring and there is no closed proper ideal I such that R = I + R0 , then R is totally disconnected.

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Theorem 3.8. If R is a compact ring, then there exists a totally disconnected compact ideal I such that R = I + R0 . Proof. Let M = {I|I is a closed subring in R and R = I + R0 }. We note that R ∈ M. Let A be a chain in M (with ⊂ as partial ordering). We claim that ∩A is the greatest lower bound for A. Indeed, let x ∈ R and A ∈ A . Since R = A + R0 , (x + R0 ) ∩ A = ∅. The family {(x + R0 ) ∩ A}A∈A is linearly ordered. Hence by compactness of R, ∩A∈A {(x+R0 )∩A} = ∅. If y ∈ ∩A∈A {(x + R0 ) ∩ A}, then y − x ∈ R0 . Thus x ∈ y + R0 ⊂ (∩A∈A A) + R0 and so R = (∩A∈A A) + R0 . By Zorn’s Lemma there exists a minimal element I in M. We claim that I is totally disconnected. Assume the contrary. By Lemma 3.6 there exists a proper closed subring J of I such that I = J + I0 . Thus R = I + R0 = J + I0 + R0 = J + R0 , a contradiction with minimality of I. By Corollary 3.7, I is totally disconnected. Since R · R0 = R0 · R = {0}, the subring I is an ideal of R.  Corollary 3.9. Let R be a compact ring. If Rn , where n ≥ 2, is open, then R is totally disconnected. Proof. By Theorem 3.8 there exists a compact totally disconnected ideal I such that R = I + R0 . We have by [14](Theorem 7.8) that R0 ⊂ Rn . Now Rn ⊂ I n ⊂ I implies R0 ⊂ Rn ⊂ I. Thus R0 = {0}.  Lemma 3.10. If (R, T ) is a vdW-ring and R is a homomorphic image of R with trivial multiplication, then R is finite. Proof. Assume the contrary. Consider on R the Bohr topology U . It is known that (R , U ) is Hausdorff (see, for instance, [10]). Since (R, T ) is a vdW-ring, (R , U ) is a continuous homomorphic image of (R, T ). By Lemma 3.1, (R , U ) is a vdW-ring. Hence every character of it is continuous. By a result of Kakutani [18] (see [14](24.47)) (R , U) has a discontinuous character, a contradiction.  Lemma 3.11. Let R be a nilpotent ring, and let R = S + R2 for some subring S. Then R = S. Proof. Induction on the index n of nilpotency. If n = 2, the assertion is obvious. Let R be a nilpotent ring of index of nilpotence n + 1. Thus R/Rn = (S + Rn )/Rn + (R2 /Rn ) = (S + Rn )/Rn + (R/Rn )2 . By induction R/Rn = (S + Rn )/Rn . Hence R = S + Rn . This implies  R2 = RS ⊂ (S + Rn )S ⊂ SS ⊂ S. One obtains R = S. Lemma 3.12. Let R = x1 , . . . , xn  be a nilpotent finitely generated ring. Then its additive group is finitely generated.

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Proof. Let m be the index of nilpotency. The set consisting of elements y1 y2 · · · yi , where i ≤ m − 1 and yj ∈ {x1 , . . . , xn }, is the set of generators of the additive group (R, +) of R.  Lemma 3.13. If R is a nilpotent ring and R2 is cofinite, then R is finite. Proof. There exists a natural number n such that nR ⊂ R2 . By induction it follows ni R ⊂ Ri+1 for all i ∈ ω. Since R is nilpotent, there is i0 ∈ ω with ni0 R = {0}. Hence the additive group of R is bounded. Lemma 3.11 and Lemma 3.12 imply that the additive group of R is finitely generated. By Theorem 2.9, S is finite.  Corollary 3.14. If R is a vdW-ring and R is a homomorphic image being a nilpotent ring, then R is finite. Proof. By Lemma 3.10, R2 is cofinite. Since R /R 2 is a homomorphic  image of R/R2 , apply Lemma 3.13 to complete the proof. Corollary 3.15. If R is a vdW-ring, then Rn is open for each n ∈ N. Lemma 3.16. Let R be a compact ring, n ∈ N, and let Rn be cofinite. Then Rn is open. Proof. There exists a finite subset F = {x1 , . . . , xs } of R such that R = F + Rn .   ∞ [n] We have that R = (∪∞ · · · (∪j=1 (xs + [j]R[n] )). i=1 (x1 + [i]R )) Since R is a Baire space, there exist i ∈ [1, s], k ∈ N such that the interior of xi + [k]R[n] is nonempty. There exists r0 ∈ xi + [k]R[n] and a neighborhood V of zero such that r0 + V ⊂ xi + [k]R[n] . This implies −xi + r0 + V ⊂ [k]R[n] ⊂ Rn . It follows that Rn is open.  Proposition 3.17. For a compact ring (R, T ) the following conditions are equivalent: (i)(R, T ) is a vdW-ring; (ii) R2 is open and every cofinite ideal of R is open. Proof. (i) ⇒ (ii): It follows from Corollary 3.15 that R2 is open. By Lemma 2.10 every cofinite ideal of R is open. (ii) ⇒ (i): Let f : (R, T ) → (R , T  ) be a surjective homomorphism, where (R , T  ) is a compact ring. Thus f (R2 ) = (R )2 is cofinite in R . According to Corollary 3.9, R is totally disconnected. If W is an open ideal of (R , T  ), then f −1 (W ) is a cofinite ideal of (R, T ), hence is open. This means that f is continuous.  Corollary 3.18. Let R be a compact ring, and let R = R1 +· · ·+Rn = {x1 + · · · + xn |xi ∈ Ri , i = 1, . . . , n}. If each Ri is a vdW-ring, then R is a vdW-ring.

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Proof. Let Ri = Fi + Ri2 , where Fi is finite, i = 1, . . . , n. Then R = F1 + · · · + Fn + R12 + · · · + Rn2 = F1 + · · · + Fn + R2 . It follows that R2 is a cofinite ideal. By Lemma 3.16, R2 is open. If V is a cofinite ideal of R, then V ∩ R1 , · · · , V ∩ Rn are closed subgroups of R. The subgroup V ∩ R1 + · · · + V ∩ Rn is closed and cofinite, hence it is open. It follows that V is open.  Corollary 3.19. If R is a compact ring and V an open vdW-subring, then R is a vdW-ring. Proof. Indeed, R2 ⊃ V 2 , hence R2 is open. If I is a cofinite ideal of R, then V ∩ I is an open cofinite ideal of V , hence I is open.  Theorem 3.20. The class of vdW-rings is closed under extensions. Proof. Let (R, T ) be a compact ring, and let I be a closed ideal such that (I, T I ) and (R/I, T /I) are vdW-rings. Let T  be the largest precompact ring topology on R and denote R = R. Since (R, T ) is compact, (R , T  ) is Hausdorff. We have to show that T  = T . Consider the (identity) map f : (R, T ) → (R , T  ) with f (r) = r for all r ∈ R. Since (I, T I ) is a vdW-ring and (R , T  ) is Hausdorff, f (I) is a compact ideal of (R , T  ). Consider the canonical homomorphism g : (R , T  ) → (R /f (I), T  /f (I)). The homomorphism h = g ◦ f : R → R /f (I) is surjective and ker h = I. Therefore, we have an isomorphism λ : R/I → R /f (I). Since (R/I, T /I) is a vdW-ring, the homomorphism λ : (R/I, T /I) → (R /f (I), T  /f (I)), x → λ(x) is continuous. Therefore, (R /f (I), T  /f (I)) is a compact ring. Since the class of compact rings is closed under extensions, (R , T  ) = (R, T  ) is a compact ring. This implies that T  = T . Second proof: Let R be a compact ring, and let I be a closed ideal of R. Assume that the rings R/I and I are vdW-rings. We claim that R2 is open. By Corollary 3.15, I 2 is open in I, and R2 + I is open. I ∩ R2 ⊃ I 2 implies that I ∩ R2 is open in I. There exists a finite subset F of I such that I = F + (I ∩ R2 ). Additionally, there exists a finite subset F1 of R such that R = F1 + R2 + I. Thus R = F1 + F + R2 + R2 ∩ I = F1 + F + R2 . Therefore, R2 is cofinite, hence by Lemma 3.16 it is open. Since R/I and I are totally disconnected by Theorem 3.4, the ring R also is totally disconnected. Thus R has a fundamental system of neighborhoods consisting of cofinite ideals. Let H be a cofinite ideal of R. The mapping λ : H → (H + I)/I, h → h + I is continuous as the restriction of the natural homomorphism of R on R/I. If V is an

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open ideal of H, then V is cofinite. Hence λ(V ) = (V + I)/I is cofinite. Since R/I is a vdW-ring, λ(V ) = (V + I)/I is open. Therefore, λ is a continuous and open homomorphism, hence H/ ker λ ∼ =top (H + I)/I. Since ker λ = H ∩ I is compact, the ring H is compact, hence it is closed. Since H is cofinite, it is open. The assertion follows from Proposition 3.17.  Corollary 3.21. If I is a closed ideal of a compact ring R and every closed ideal of I and every closed ideal of R/I are vdW-rings, then every closed ideal of R is a vdW-ring. Proof. Let L be a closed ideal of R. Then L∩I is a closed ideal of R and hence also of I. Thus L∩I is a vdW-ring. It is L/(L∩I) ∼ =top (L+I)/I, and (L + I)/I is a closed ideal of R/I. Then L/(L ∩ I) is a vdW-ring, too. Now apply Theorem 3.20 to complete the proof.  Lemma 3.22. Let A be a compact Abelian torsion group which is strongly complete. Then A is finite.  Proof. By [14](25.9) A is topologically isomorphic to i∈I Z(bi ), where there are only finitely many positive integers bi and I is an arbitrary nonvoid index set. Assume that A is infinite. Then there is i0 ∈ I such that Z(bi0 )ω is a direct factor of A. By [28](Example 4.2.12, p. 12) this factor is not strongly complete. Since a quotient of a strongly complete group is strongly complete – see [28](Exercise 4.2.14(a)), A is not strongly complete, a contradiction.  Theorem 3.23. Every open subring S of a vdW-ring R is a vdW-ring. Proof. Every cofinite ideal I of S is a cofinite subring of R. Since R is a vdW-ring, Lemma 2.6 implies that I is open in R. By Proposition 3.17 we have to show that S 2 is an open subring of S. Case I. The additive group of R is bounded. The group S/S 2 is bounded and by Theorem 2.9 it is a direct sum of cyclic groups Cα (α ∈ Ω). Denote by ϕ the canonical homomorphism of S on S/S 2 . Furthermore, denote for each α ∈ Ω by ϕα the projection of S/S 2 on Cα and put Hα = ker(ϕα ◦ ϕ). Then Hα is a cofinite subgroup of S and S 2 ⊂ Hα (α ∈ Ω), hence S 2 ⊂ ∩α∈Ω Hα . If x ∈ ∩α∈Ω Hα , then (ϕα ◦ ϕ)(x) = 0 for all α ∈ Ω. Hence ϕ(x) = 0, i.e., x ∈ S 2 . We obtain that S 2 = ∩α∈Ω Hα , where each Hα is a cofinite subgroup of S. Since each Hα is a cofinite subring of R, it is open. It follows that S 2 is closed.

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Every cofinite subgroup of S/S 2 has the form H/S 2 , where H is a cofinite ideal of S. Hence it is open. Thus S/S 2 is an Abelian compact bounded group all whose cofinite subgroups are open. It follows from Lemma 3.22 that S/S 2 is finite. Hence S 2 is open. Case II. R is arbitrary. Since S is cofinite, there exists n ∈ N such that nR ⊂ S. This implies n2 R2 ⊂ S 2 . Since R is a vdW-ring, R2 is open by Corollary 3.15. Hence there exists k ∈ N such that kR ⊂ R2 . This implies that n2 kR ⊂ S 2 . Since R is compact, n2 kR is a closed ideal. The factor ring R/n2 kR is a vdW-ring and S/n2 kR is a cofinite subring. By Case I, (S/n2 kR)2 = (S 2 + n2 kR)/n2 kR = S 2 /n2 kR is  open. Thus S 2 is open. Theorem 3.24. A compact ring R is a vdW-ring if and only if for every cofinite ideal I the ideal I 2 is open. Proof. ⇒: Every cofinite ideal I of R is open. Hence I is a vdW-ring by Theorem 3.23. Now Corollary 3.15 implies that I 2 is open. ⇐: Since R is a cofinite ideal, R2 is open. If I is a cofinite ideal, then I 2 ⊂ I is open, hence I is open. Now apply Proposition 3.17  Theorem 3.25. Let R be an arbitrary van der Waerden ring and n ∈ N. Then the ring M (n, R) with the standard topology is a vdW-ring. Proof. Induction on n. For n = 1, M (n, R) = R and the theorem is trivial. Assume that the theorem was proved for n − 1. We will prove it for n. Let f : M (n, R) → R be a homomorphism of M (n, R) into a precompact ring R . We will show that f is continuous. The subring A of M (n, R) consisting of matrices of the form ⎡ ⎤ r11 r12 · · · r1n−1 0 ⎢ r21 r22 · · · r2n−1 0 ⎥ ⎢ ⎥ ⎢ ··· ··· ··· ··· ··· ⎥ ⎢ ⎥ ⎣ rn−11 rn−12 · · · rn−1n−1 0 ⎦ 0 0 ··· 0 0 is topologically isomorphic to M (n − 1, R). Analogously, the subring B consisting of matrices of the form ⎡ ⎤ 0 0 ··· 0 0 ⎢ 0 r2n ⎥ r22 · · · r2n−1 ⎢ ⎥ ⎢ ··· ··· ··· ⎥ · · · · · · ⎢ ⎥ ⎣ 0 rn−12 · · · rn−1n−1 rn−1n ⎦ 0 rn2 · · · rn−1n rnn

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is topologically isomorphic to M (n − 1, R). We notice that the subring C of M (n, R) consisting of matrices of the form ⎡ ⎤ r r ··· r r ⎢ 0 0 ··· 0 0 ⎥ ⎢ ⎥ ⎣ ··· ··· ··· ··· ··· ⎦ 0 0 0 0 0 in the induced is topologically isomorphic to R → C, ⎡ r r ··· r ⎢ 0 0 ··· 0 r → ⎢ ⎣ ··· ··· ··· ··· 0 0 ··· 0

R. Indeed, the mapping ⎤ r 0 ⎥ ⎥ ··· ⎦ 0

is a topological isomorphism. In an analogous way the subring D of M (n, R) consisting of matrices of the form ⎡ ⎤ r 0 ··· 0 0 ⎢ r 0 ··· 0 0 ⎥ ⎢ ⎥ ⎣ ··· ··· ··· ··· ··· ⎦ r 0 ··· 0 0 is topologically isomorphic to R. Let now V  be any neighborhood of zero of R . Choose a symmetric neighborhood U  of zero of R such that U  +U  +U  +U  +U  +U  ⊂ V  . Since the subring B is a vdW-ring, there exists a neighborhood U1 of zero of R such that ⎡ ⎤ 0 0 ··· 0 0 ⎢ 0 U 1 · · · U1 U 1 ⎥  ⎥ f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ ⊂ U . 0 U1 · · · U1 U1 Analogously there exists a neighborhood U2 ⊂ U1 of zero of R such that ⎤ ⎡ U2 U2 · · · U2 0 ⎢ U2 U2 · · · U2 0 ⎥ ⎥ ⎢ ⎥ ⊂ U . · · · · · · · · · · · · · · · f⎢ ⎥ ⎢ ⎣ U2 U2 · · · U2 0 ⎦ 0 0 ··· 0 0

VAN DER WAERDEN RINGS

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Since C is a vdW-ring, there exists a neighborhood U3 ⊂ U2 of zero of R such that f [M (n, U3 ) ∩ C] ⊂ U  . We claim that ⎤ ⎡ 0 0 · · · 0 U3 ⎢ 0 0 ··· 0 0 ⎥   ⎥ f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ ⊂ U + U . 0 0 ··· 0 0 Indeed, if u ∈ U3 , then ⎤ ⎡ 0 0 ··· 0 u ⎢ 0 0 ··· 0 0 ⎥ ⎥ f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ 0 0 ··· 0 0 ⎡ ⎤ u u ··· u u ⎢ 0 0 ··· 0 0 ⎥ ⎥ =f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ 0 0 ··· 0 0 ⎡ ⎤ u u ··· u 0 ⎢ 0 0 ··· 0 0 ⎥ ⎥ −f ⎢ ⎣ ··· ··· ··· ··· ··· ⎦ 0 0 ··· 0 0 ⎤ ⎡ U2 U2 · · · U2 0 ⎢ U2 U2 · · · U2 0 ⎥ ⎥ ⎢   ⎥ ∈ f [M (n, U3 ) ∩ C] − f ⎢ ⎢ ··· ··· ··· ··· ··· ⎥ ⊂ U + U . ⎣ U2 U2 · · · U2 0 ⎦ 0 0 ··· 0 0 In analogous way there exists a neighborhood U4 ⊂ U3 of zero of R such that ⎡ ⎤ 0 0 ··· 0 0 ⎢ 0 0 ··· 0 0 ⎥   ⎥ f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ ⊂ U + U . U4 0 · · · 0 0 It follows immediately that ⎡ ⎤ U4 U4 · · · U4 U4 ⎢ U4 U4 · · · U4 U4 ⎥ ⎥ f⎢ ⎣ ··· ··· ··· ··· ··· ⎦ U4 U4 · · · U4 U4

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⊂ U  + U  + U  + U  + U  + U  ⊂ V .



Remark 3.26. It can be proved an analogous result for the ring T (n, R) of upper triangular matrices over a vdW-ring R. Remark 3.27. We give two examples of a vdW-ring whose center is not a vdW-ring. Example (a) shows also that the intersection of a family of vdW-subrings of a vdW-ring is not necessarily a vdW-ring. In example (b) the vdW-ring is a prime ring.  (a) Let p be a prime number and n ∈ N. Consider the product R = n∈N M (n, Fp ) of the discrete finite rings M (n, Fp ). By Theorem 2.8, R is a vdW-ring. Its center Z(R) is topologically isomorphic to Fℵp 0 which is not a vdW-ring by Remark 5.21. If {Vn }n∈N is a fundamental system of zero of R consisting of open ideals, then Z(R) = ∩n∈N [Z(R) + Vn ]. By Theorem 3.23 each subring Z(R) + Vn is a vdW-ring, but Z(R) not. (b) Let p be a prime number, and let R = Fp x, y be the ring of commutative formal power series with the usual compact topology. Then R is a local compact ring with radical topology. By Lemma 5.2, R is a vdW-ring. i The ideal Rx = Σ∞ i=1 Fp yx is a closed compact radical ring. We notice that Rx is not a vdW-ring. Indeed, (Rx)2 = Rx2 = i 2 Σ∞ i=2 Fp yx , and the factor group Rx/(Rx) is topologically isomorphic to Fp yx. Hence it is infinite. By Lemma 3.10, Rx is not a vdW-ring. Rx is an open subring of the compact ring Fp + Rx. It follows from Theorem 3.23 that Fp + Rx of R is not a vdW-ring. The following ring S goes back to Small [30]:  R Rx . S= R Fp + Rx Since S is a closed subring of M (2, R), it is compact. Straithforward calculations show that Z(S) is topologically isomorphic to Fp + Rx.      x 0 y 0 0 x 0 0 0 0 The elements , , , , are 0 0 0 0 0 0 1 0 0 1 topological generators of S, but Z(S) is not finitely generated. Direct calculations show that S is a prime ring. 4. Metrizability of van der Waerden Rings It was noted in [7](p. 22) that the metrizability of vdW-rings follows from [34]. We give here a new proof based on different arguments.

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We will use methods of varieties of rings (see [8](Chapter IV) or [24](Chapter V). Notation. If R is a ring, then MR stands for the variety of rings generated by R. Lemma 4.1. Let R be a finite ring and S be a semisimple vdW-ring belonging MR . Then S is finite. Proof.  Assume on the contrary that S is infinite. Theorem 2.4 implies S = α∈Ω M (nα , Fα ), where each Fα is a Galois field and nα ∈ N. Since R is finite, there exist n, k ∈ N, n > k such that R satisfies the identity xn = xk . If F ∈ MR is a field, then xn−k = 1 for all 0 = x ∈ F . This implies that |F | ≤ n − k + 1. If P ∈ MR , then the index of nilpotence of each nilpotent element a ∈ P is ≤ n − k. If L is a finite ring with identity and k ∈ N, then the ring M (k, L) contains a nilpotent element of index of nilpotence equal to k. It follows that the numbers nα , α ∈ Ω are bounded and that the cardinalities kα , α ∈ Ω are bounded. Then S contains a ring M (s, F )ω as a topological direct summand, where F is a finite field and s ∈ N. It follows that M (s, F )ω is a vdW-ring. The ring M (s, F )ω is topologically isomorphic to M (s, F ω ). If I is a maximal ideal of F ω containing ⊕ω F , then I is nonopen and cofinite. Thus the ideal M (s, I) will be a nonopen cofinite ideal, a contradiction.  Lemma 4.2. If R is a compact nilpotent ring with bounded additive group all whose cofinite ideals are open, then R is finite. Proof. By Lemma 3.22, R/R2 is finite. Then there exists a finite subring S such that R = S + R2 . Since R is totally disconnected, it is linear. If V is an open ideal of R, then R = S + V + R2 . Since R is nilpotent, Lemma 3.11 implies R = S + V . Then R = S, because S is finite.  Lemma 4.3. Let R be a vdW-ring, and let R be algebraically and topologically embedded in K ω , where K is a finite ring. Then R is finite. Proof. By Lemma 4.1, R/J(R) is finite. Moreover, J(R) is a compact nilpotent ring and its additive group is bounded. We claim that every cofinite ideal in J(R) is open. Indeed, every cofinite ideal of J(R) is a cofinite subring of R. Since every cofinite ideal in R is open, Lemma 2.6 implies that J(R) has the same property. By Lemma 4.2, J(R) is finite, hence R is finite.  Theorem 4.4. Every vdW-ring is of type (AF).

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Proof. By Theorem 3.4, R is a linear ring. Assume the contrary. Then there exist k ∈ N and a family {Vn }n∈N of different open ideals of index k. Let I = ∩n∈N Vn . If s = t, then the images of Vs and Vt in R/I are different. It follows that R/I is an infinite vdW-ring. There exists a finite ring K such that R/I is algebraically and topologically embedded in K ω . A contradiction with Lemma 4.3.  Corollary 4.5. Every vdW-ring is metrizable. 5. Structure Theorems for van der Waerden Rings Lemma 5.1. If R is a vdW-ring with open radical and J(R)2 = {0}, then R is finite. Proof. Since J(R) is cofinite, by Theorem 3.24, {0} = J(R)2 is open, hence R is discrete and so R is finite.  Lemma 5.2. Any compact ring R with radical topology is a vdW-ring. Proof. Let I be a cofinite ideal. Then there exists n ∈ N such that J(R)n ⊂ I. Thus J(R)2n ⊂ I 2 . Hence I 2 is open. By Theorem 3.24, R is a vdW-ring.  By [32](Chapter 2, Corollary 6.45) every compact left Noetherian ring R has radical topology. Now Lemma 5.2 implies that R is a vdWring. This generalizes [7](Corollary 4.6). Problem 5.3. Let R be a compact left Noetherian ring. Is Z(R) Noetherian, too? We will say that a topological ring is a p-ring (p ∈ P) provided lim pn x = 0 for all x ∈ R. These rings are called also p-primary (see

n→∞

[36](p. 334)). Proposition 5.4. Let R be a compact ring with open radical and as2 . Then R is totally sume that there exists n ∈ N such that nR ⊂ J(R)  disconnected and topologically isomorphic to p∈P0 Rp , where P0 ⊂ P is a finite subset of P and each Rp is a p-ring. Proof. By induction ni J(R) ⊂ J(R)i+1 for every i ∈ N. Therefore, each quotient ring J(R)/J(R)i has a torsion additive group and so R/J(R)i is linear. Since R ∼ =top limR/J(R)i , R is linear. Set k = [R : J(R)]. ←

Then each ring R/J(R)i has a finite characteristic mi and all prime factors in the decomposition of mi are divisors of k or n.

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Now let P0 = {p ∈ P|p divides kn}. By a theorem of Jacobson ([33], Theorem I.I.34, p. 14), R ∼ =top p∈P Rp , where Rp = {x ∈ m R| lim p x = 0}. We identify R with p∈P Rp . We obtain that for m→∞   each i ∈ N the image of p∈P0 {0}p × q∈P / 0 Rq under the canonical homomorphism of R on R/J(R)i is zero. It follows immediately that   R = p∈P0 Rp . Theorem 5.5. A compact ring R with open radical is a vdW-ring if and only if R is a ring with radical topology. Proof. ⇒: According to Lemma 3.1, R/J(R)2 is a vdW-ring. Evidently, J(R/J(R)2 )2 = {0}. By Lemma 5.1, R/J(R)2 is finite, and by Proposition 5.4, R is totally disconnected. Then by [32](Chapter 2, Theorem 6.43) R is a ring with radical topology. ⇐: Follows from Lemma 5.2.  Theorem 3.20 and Theorem 5.5 imply Corollary 5.6. A compact ring R is a vdW-ring whose Jacobson radical is a vdW-ring if and only if R/J(R) is a vdW-ring and J(R) is a ring with radical topology. Corollary 5.7. A vdW-ring R with open radical is topologically finitely generated. Proof. Follows from Theorem 5.5 and [32](Chapter 2, Theorem 6.43).  Corollary 5.8. A compact integral domain is a vdW-ring if and only if its topology is radical. By the proof of [17](Theorem 19) the radical of a compact integral domain is open. Remark 5.9. For every p ∈ P the additive group (Zp , +) of the ring Zp of p-adic integers considered as a ring with trivial multiplication has a finite number of topological generators and its radical is open, but is not a vdW-ring. Remark 5.10. If R is a topologically finitely generated topological ring and S is an open cofinite ring, then S is topologically finitely generated: Indeed, let P be a dense finitely generated ring of R. Then P ∩ S is a cofinite subring in P . Therefore, by Theorem 2.7 it is finitely generated. Since S = S ∩ P , the proof is complete.

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Lemma 5.11. If R is a compact topologically finitely generated ring with identity and open radical, then its topology is radical. Proof. By [32](Chapter 2, Theorem 6.43) it suffices to show that J(R)2 is cofinite. The quotient ring S = R/J(R)2 is topologically finitely generated and its radical J(S) = J(R/J(R)2 ) = J(R)/J(R)2 is open. If nS ⊆ J(S), then n2 S 2 = n2 S = {0}, i.e., S has a finite characteristic. Hence J(S) is an Abelian bounded group. By Remark 5.10 it is topologically finitely generated as a compact Abelian group, since J(S) has the zero-multiplication. It follows that J(S) is finite and so S is finite.  Theorem 5.12. A compact ring R with identity and open radical is a vdW-ring if and only if it is topologically finitely generated. Proof. ⇒: That is Corollary 5.7. ⇐: Follows from Lemma 5.11 and Theorem 5.5.



Corollary 5.13. A compact integral domain with identity is a vdWring if and only if it is topologically finitely generated. Theorem 5.14. Every vdW-ring R can be embedded into a vdW-ring with identity. If R is a vdW-ring of characteristic n, then it can be embedded as an open ideal into a vdW-ring of characteristic n and with identity. Proof. According to Theorem 3.4, R is a linear ring. Then by [33](pp. 17-18] R can be embedded into a compact ring R with identity e. We consider R = e, R = e + R. It suffices to show that every cofinite ideal I of R is open. We notice that I ∩ e is a cofinite ideal of e. By [32](Chapter 2, Theorem 6.34) I ∩ e is open, hence closed in e. The ideal I ∩ R is closed in R, hence it is closed in R . The subgroup H = I ∩ e + I ∩ R of R is contained in I and is closed. We claim that H is cofinite in R . Indeed, there exist a finite subset F of e such that e = F + I ∩ e and a finite subset G of R such that R = G+I ∩R. Then R = e+R = F + I ∩ e + G + I ∩ R = F + G + I ∩ e + I ∩ R = F + G + H. Then H is open, and so I is open. Assume now that charR = n. We embed R into a ring R with identity in the usual way: consider the topological product R = Z/nZ × R, where the multiplication is given by the rule (α, r)(β, s) = (αβ, αs + βr+rs), α, β ∈ Z/nZ, r, s ∈ R. Then R becomes a compact topological  ring. By the same arguments R is a vdW-ring.

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Lemma 5.15. Let (R, T0 ) be a compact ring, and let R = R1 ⊕· · ·⊕Rn be a decomposition of (R, T0 ) into a direct topological sum of its ideals R1 , · · · , Rn . Then (R, T0 ) is a van der Waerden ring if and only if each (Ri , T0 Ri ) is a vdW-ring. Proof. ⇒: Follows from Lemma 3.1. ⇐: Follows from Corollary 3.18.



Theorem  5.16. An Abelian compact ring R a vdW-ring if and only if R∼ =top ( i∈I Ri ) × L, where: (i) each Ri is a local topologically finitely generated ring and for each i0 ∈ I the set of j ∈ I such that the fields Ri0 /J(Ri0 ) and Rj /J(Rj ) are isomorphic, is finite, (ii) L is a radical ring with radical topology. Proof. ⇒:(i): If R is a radical ring, apply Theorem 5.5. Now assume J(R) = R. Consider the quotient ring R/J(R), and let e + J(R) be the identity of R/J(R). Then there exists an idempotent e ∈ e + J(R) by Theorem 2.11. By assumption e is central. Let R = Re ⊕ R(1 − e), where R(1 − e) = {x ∈ R|xe = 0} is the Peirce decomposittion of R with respect to the idempotent e (see section 2). R is the topological sum of the ideals A = Re and L = R(1 − e), where A has an identity. According to [32](Chapter 2, Theorem 18.3) A is isomorphic to a topo logical product i∈I Ri of compact local rings. By Lemma 3.1 each Ri is a vdW-ring. According to Corollary 5.7 each Ri is topologically finitely generated. By Lemma 3.1 and [36]((26.16), (26.21)) the quotient  ring R/J(R) is a semisimple vdW-ring topologically isomorphic to i∈I Ri /J(Ri ). By Theorem 2.8, for each i0 ∈ I the set of j ∈ I such that fields Ri0 /J(Ri0 ) and Rj /J(Rj ) are isomorphic is finite. (ii): Let ϕ be the canonical homomorphism of R onto R/J(R). Then ϕ(L) = {0}, hence L ⊂ J(R). Since an ideal of a radical ring is radical, L is a compact radical ring. By Theorem 5.5, L is endowed with the radical topology. ⇐: By Theorem 5.5, L is a vdW-ring with  radical topology. Then by Lemma 5.15 it suffices to show that A = i∈I Ri is a vdW-ring. Since A has identity, it suffices to show by Lemma 2.10 that each  cofinite ideal  ∼ K of A is open. By Theorem 2.8, A/ J(R ) = i top i∈I  i∈I Ri /J(Ri ) is  J(R )/ J(R ) is an open ideal a vdW-ring. Thus the ideal K+ i i i∈I i∈I  of A/ i∈I J(Ri ). Denote  for each i ∈ I by ei the identity of Ri and by fi the idempotent ei × j=i 0j . The family {fi |i ∈ I} of A is summable  and its sum g = i∈I ei is theidentity of A. Denote by φ the canonical homomorphism of A on A/ i∈I J(Ri ). The family {φ(fi )|i ∈ I} of

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i∈I J(Ri ) is summable, and its sumis equal to φ(g).Since (K + J(R ))/ i i∈I i∈I J(Ri ) is open in A/ i∈I J(Ri ), K + i∈I J(Ri ) is open in A. Thisimplies that there  exists a finite subset I0 of Isuch that f = i∈I0 0i × j ∈I e ∈ K + j / 0 i∈I J(Ri ). There exists a ∈ i∈I J(Ri ) such that f + a ∈ K. Then f + f a ∈ K. If b is the quasiinverse of a, i.e., a + b + ab = 0, then (f + f a)(f + f b) = f + f (a + b + ab) = f ∈ K. It follows that f K ⊂ f A ⊂ f K, hence f A = f  K. We have K = (g − f )K ⊕ f K = (g − f )K ⊕ ( j ∈I a topological / 0 Rj ),  direct sum of ideals. Thus (g − f )K is a cofinite ideal of i∈I0 Ri . By Theorem 5.12 each Ri , i ∈ I is a vdW-ring. According to Lemma 5.15 Therefore (g − f )K is an open ideal, the ring i∈I0 Ri is a vdW-ring.  hence a compact ideal of i∈I0 Ri . This implies that K is a closed ideal of A, hence K is open. We proved that A is a vdW-ring. 

Remarks 5.17. Let L and R be defined as in Theorem 5.16. (a) The ring L is topologically isomorphic to a finite product of p-rings with the radical topology. (b) L = {0} if and only if R has an identity. R is a radical ring if and only if I = ∅. (c) By the help of Theorem 2.11 it is easy to see that R/J(R) is commutative, since R is Abelian. Thus Theorem 2.8 is not contained in Theorem 5.16 for non-commutative semisimple vdW-rings. A ring R is said to be reduced if R has no nonzero nilpotent elements, or, equivalently, if x2 = 0 implies x = 0 in R ([21], p. 4). Clearly, products of integral domains are reduced. Note that in a reduced ring R all idempotents are central: Indeed, let x be an arbitrary element of R. Consider the element y = ex−exe. Then y 2 = (ex−exe)(ex−exe) = exex − exex − exexe + exexe = 0. This implies y = 0 which means ex = exe. If we consider the element z = xe − exe, then z 2 = 0, hence z = 0, i.e., xe = exe and so xe = ex. Corollary5.18. A compact, reduced ring R is a vdW-ring if and only if R ∼ =top ( i∈I Ri ) × L, where: (i) each Ri is a local topologically finitely generated reduced ring and for each i0 ∈ I the set of j ∈ I such that fields Ri0 /J(Ri0 ) and Rj /J(Rj ) are isomorphic, is finite, (ii) L is a radical reduced ring with radical topology. Corollary 5.19.  A compact commutative ring R is a vdW-ring if and only if R ∼ =top ( i∈I Ri ) × L, where: (i) each Ri is a compact local Noetherian ring and for each i0 ∈ I the set of j ∈ I such that fields Ri0 /J(Ri0 ) and Rj /J(Rj ) are isomorphic, is finite,

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(ii) L is a commutative radical ring with radical topology. Proof. ⇒: Keep the notation of Theorem 5.16. Each Ri is a compact local ring whose topology is radical. Then by [26](Theorem 12) and [33](Theorem 1.10.12) each Ri is Noetherian. ⇐: Follows from Theorem 5.16, since by [32](Chapter 2, Corollary 6.45) every compact (left) Noetherian ring is topologically finitely generated.  Theorem 5.20. Let (R, T ) be a compact ring with identity. Then R is a vdW-ring if and only if (R[[X]], T0 ) (the construction of the topology T0 is indicated in section 2) is a vdW-ring. Proof. ⇐: Follows from Lemma 3.1, since (R, T ) is a continuous homomorphic image of (R[[X]], T0 ). ⇒: By Lemma 2.10 we have to show that every cofinite ideal I of (R[[X]], T0 ) is open. The ideal I ∩ R of R is cofinite. Hence it is open in (R, T ). The ideal XR[[X]] of (R[[X]], T0 ) is closed and topologically nilpotent. Thus it is a radical ring. The image S of XR[[X]] in R[[X]]/I under the canonical homomorphism is a finite radical ring. Hence it is nilpotent. There exists n ∈ N such that S n ⊂ I. In particular, X n ∈ I. This implies that X n R[[X]] ⊂ I. Then R ∩ I + (R ∩ I)X + · · · + (R ∩ I)X n−1 + X n R[[X]] ⊂ I. Hence I is open  in (R[[X]], T0 ). Remark 5.21. (Compare with [28](Example 4.2.13)) Let I be an infinite set, and let R be a fixed nonzero finite ring. Then the ring S = RI contains a nonopen ideal H of index |R|. The ring S is not a vdW-ring. Let F be an ultrafilter on I containing the filter of all cofinite subsets of I, and let H be the collection of all elements h = (hi ) ∈ S such that {i ∈ I|hi = 0} ∈ F. Evidently, H is a proper ideal of S. Since F contains all cofinite subsets of I, H is dense. For each r ∈ R, define r ∈ S the element all whose components ri are equal to r. Obviously, if r = s, then r − s ∈ / H. To see that [S : H] = |R|, it suffices to show that every element x ∈ S is congruent to some r modulo H. Fix x ∈ S; for r ∈ R define Ir = {i ∈ I|xi = r}. Then I = ∪r∈R Ir . Since F is an ultrafilter, Ir ∈ F for some r ∈ R. Then x − r ∈ H. The ideal H is not open, since it is proper of finite index and it is dense. This construction was given first for finite groups in [27]. Example 5.22. A commutative vdW-ring whose radical is not a vdWring:

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Consider the Galois fields F2n , n ∈ N, and let Rn be the algebra over 2 F2n generated by  1 and one element an with an = 0. By Theorem 5.16 the ring R = n∈N Rn is a vdW-ring. Its radical J(R) has trivial multiplication and is infinite. Hence is not a vdW-ring by Lemma 3.10. This example shows that the intersection of vdW-ideals is not necessarily a vdW-ideal. (A closed ideal of a compact ring is called a vdW-ideal if it is a vdW-ring.) The radical J(R) is closed, and the compact ring R is linear. Therefore, J(R) is the intersection of all open ideals containing it. By Theorem 3.23 each of those ideals is a vdW-ideal, but J(R) is not a vdW-ideal. Problem 5.23. Is it true that the intersection of two vdW-ideals of a compact ring is a vdW-ideal? Proposition 5.24. Let {Rα |α ∈ Ω} be a family of rings withidentity, subset of Ω. IfI is an ideal ofR = α∈Ω Rα , and let Ω0be a finite  then I ∩ ( α∈Ω0 Rα × β ∈Ω / 0 {0}β ) = α∈Ω0 prα (I) × β ∈Ω / 0 {0}β . Proof. Denote for each α ∈ Ω by eα the identity of Rα .   The inclusion ” ⊂ ” is evidently. Conversely, let x = α∈Ω0 yα × Ω . Then β ∈Ω / 0 0β ∈ I, where yα ∈ prα (I) for each α ∈ Ω0 . Fix α ∈  0 there exists zα ∈ I such that prα (zα ) = yα . Thus zα (eα × β=α 0β ) =    yα × β=α 0β ∈ I. This implies x ∈ I ∩ ( α∈Ω0 Rα × β ∈Ω / 0 {0}β ).  Lemma 5.25. For each n, k ∈ N and for every p ∈ P the ring M (n, Fpk ) contains a subring P such that 1 ∈ P and P ∼ = Fpkn . Proof. It is well known that the field Fpkn contains an isomorphic copy of Fpk ([22](Theorem 5.5)). Denote V = Fpkn and consider V as a vector Fpk -space. Consider the regular representation of Fpkn : Fpkn → End V, α → ψ(α) = Lα , where Lα : V → V, v → αv. The image ψ(Fpkn ) of Fpkn is a subring of End V containing 1 and isomorphic to Fpkn . Since dimFpk V = n, End V is isomorphic to M (n, Fpk ).  2

Lemma 5.26. If R is a finite simple ring, k ∈ N and k k < |R|, then R contains a subring S with 1 ∈ S such that S is a field with |S| > k. Proof. Let R = M (n, Fpt ). If p = charR > k, the Lemma is obvious. 2 2 2 Assume that p ≤ k. Then pk ≤ k k < ptn ⇒ k 2 < tn2 < t2 n2 ⇒ k < tn. According to Lemma 5.25 R contains a subring S containing  1 and being a field with ptn > nt > k elements. Lemma 5.27. Let S be a finite semigroup. Then there exist s, k ∈ N with k > s such that ∀x∈S [xs = xk ]

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Proof. By [32](Chapter 2, Proposition 4.19) every compact semigroup contains an idempotent. Let S = {x1 , · · · , xn }. There exist natural i are idempotents for i = 1, · · · , n. numbers m1 , · · · , mn such that xm i lmi mi Then xi = xi for all l ∈ N . In particular, we can take l > 1. We 1 ···mn 1 ···mn = xm . Put k = lm1 · · · mn and s = m1 · · · mn . obtain that xlm i i  Then k > s and xk = xs for all x ∈ S. We fix a faithful indexing {Rn }n∈ω of all finite nonisomorphic associative simple rings. We can assume without loss of generality that |R0 | ≤ |R1 | ≤ · · · . It is obvious that lim |Ri | = ∞. i→∞

sequence of elements from ω. If I Lemma 5.28. Let n0 , n1 , · · · be a  is a cofinite ideal of the ring R = i∈ω Rini , then there exists a finite   nj subset K of ω such that I = i∈K pri (I) × j∈K / Rj . Proof. We claim  thatnjthere exists i0 ∈ ω such that  i≤i0 {0}i × j>i0 Rj ⊂ I. By Lemma 5.27 the quotient ring R/I satisfies an identity of the form xk = xs , where k, s ∈ N, k > s. Passing to inverse images, we obtain that ∀x∈R [xk − xs ∈ I]. n Every ring Rj j , j ∈ ω is zero or contains a subring with 1 isomorphic to Rj . Since lim |Ri | = ∞, we can select i0 ∈ N such that | Ri |> 2

i→∞

(k + 1)(k+1) for i ≥ i0 . We identify for each j ∈ ω the ring Rj with its n diagonal image in Rj j in the case when nj > 0 (recall that if nj = 0, n then we put Rj j = 0). Fix according Lemma 5.26 for each i ≥ i0 a subring Si of Ri with identity which is a field with | Si |> k +1. Denote by θi , i ≥ i0 a generator of the multiplicative group of the field Si and ni . by ei the identity  of Ri  We claim that i≤i0 0i × j>i0 ej ∈ I. Indeed, consider the element x =   , where εj = 1 if nj > 0 and εj = 0 if nj = 0. Then i≤i0 0i ×  j>i0 εj θj  k s x − x = i≤i0 0i × j>i0 ((εj θj )k − (εj θj )s ) ∈ I. If nj > 0, then by j > n i0 , (εj θj )k − (εj θj )s is an invertible element of Rj j . Set yj = (εj θj )k −     (εj θj )s for j > i0 . Then ( i≤i0 0i × j>i0 yj )( i≤i0 0i × j>i0 yj−1 ) =     nj i× j ⊂ I. Since i≤i0 0i × j>i0 ej ∈ I. This implies i≤i0 {0} j>i0 R  the lattice of ideals of R is modular, I = I ∩ ( i≤i0 Rini × j>i0 {0}j ) +   n ( i≤i0 {0}i × j>i0 Rj j ). Since each Rini is a ring with identity, by   n Proposition 5.24, I = i≤i0 pri (I) × j>i0 Rj j .  Now we can give a new proof of Theorem 2.8.  Proof. ⇒: By Theorem 2.4, R has the form i∈ω Rini and n0 , n1 , . . . is a sequence of cardinal numbers. Lemma 3.1 implies that each Rini is a vdW-ring. By Lemma 4.1 each ni is finite.

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⇐: Follows from Lemma 5.28.



Lemma 5.29. Let m0 , . . . , ms ∈ ω, k ∈ N, and let P0 , · · · , Ps be pairwise nonisomorphic finite simple rings. Consider the ring R = P0m0 × · · · × Psms and assume that Pimi is generated by the set (i) (i) {x1 , . . . , xk }, i = 0, . . . , s. Then R is generated by the set (0) (s) {y1 , . . . , yk }, where yj = (xj , . . . , xj ), j = 1, . . . , k. Proof. We can assume without loss of generality that mi ∈ N, i = 0, . . . , s. Set S = yj |j ∈ [1, k] and denote by πi , i ∈ [0, s] the projection of R on Pimi . Claim 1. For every i ∈ [0, s], S/(S ∩ ker πi ) ∼ = Pimi . Indeed, πi (S) = m i R,. Hence S/ ker(πi S ) ∼ = Pi . Since ker(πi S ) = S ∩ ker πi , S/(S ∩ mi ∼ ker πi ) = Pi . Denote for each i ∈ [0, s], Ii = S ∩ ker πi . Claim 2. For i = j, Ii + Ij = S. Indeed, Ii + Ij ⊃ Ii (Ii + Ij ⊃ Ii ). Hence S/(Ii + Ij ) is a homomorphic image of S/Ii (S/Ij ). Then l S/(Ii +Ij ) is isomorphic to a ring of the form Pini , 0 ≤ ni ≤ mi (Pj j , 0 ≤ lj ≤ mj ). Since Pi and Pj are nonisomorphic finite simple rings, S/(Ii + Ij ) = {0}, i.e., S = Ii + Ij . We proved that the ideals Ii , i ∈ [0, s] are comaximal (i.e., Ii +Ij = S for i = j) and S/I with identity. By [16](Propositions 1,2, is are rings ∼ ∼ Chapter 1), S = i=0 S/Ii = R. Thus S = R.  Now we use the indexing indicated before Lemma 5.28. Denote, for each n ∈ N, by Mn the variety of rings generated by the ring Rn . According to [8](Corollary 3.14) the variety Mn is locally finite (i.e., every finitely generated ring in Mn is finite). Fix k ∈ N. We will associate to each n ∈ N in a unique way another number kn ∈ N : Let Sn be the free k-generated ring of the variety Mn . Since Sn is finite and has prime characteristic by the Wedderburn-Mal’cev Theorem (see, e.g., [9] or [32]), there exists a semisimple subring Qn such that Sn = Qn + J(Bn ), Sn ∩ J(Bn ) = {0}. Since Qn ∼ = Sn /J(Bn ), the ring Qn is unique up to an isomorphism. Denote by kn the number of factors in the decomposition of Qn which are isomorphic to Rn . Obviously, all kn are defined correctly. Assume now that we have a ring isomorphic to Rnm , m ∈ N and that this ring is k-generated. There exists a homomorphism f of Sn on Rnm . We have that f (Qn ) = Rnm . The simple factors of Qn nonisomorphic to Rn are contained in ker f . Therefore Rnm is a homomorphic image of Rnkn . We have proved the following Lemma. Lemma 5.30. Let n ∈ ω, and let k ∈ N. Then there exists kn ∈ N such that Rnkn is k-generated and each k-generated ring of the form Rnm , m ∈ ω is a homomorphic image of Rnkn (in particular, m ≤ kn ).

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Theorem 5.31. Let k ∈ N. Then a compact semisimple ring R is k generated if and only if R has the form n∈ω Rnln , 0 ≤ ln ≤ kn for each n ∈ ω.  Proof. ⇒: By Theorem 2.4 we can consider that R = n∈ω Rntn , where tn are cardinal numbers. Each ring Rntn is topologically k-generated and belongs to the variety Mn generated by Rn . It follows that Rntn is finite and by Lemma  5.30, tn ≤ kn . ⇐: Let R = n∈ω Rnln , 0 ≤ ln ≤ kn . We will write elements of R in the form x = (x0 , x1 , . . .), xi ∈ Rili , i ∈ ω. Choose for each i ∈ ω (i) (i) (0) (1) a set {y1 , . . . , yk } of generators of Riki . Put zj = (yj , yj , . . .) ∈ R for each j ∈ [1, k]. Denote by P the subring of R generated by the set {z1 , . . . , zk }. Denote by πs , s ∈ ω the projection of R on R0l0 × · · · × (0) (s) Rsls . Then for every j ∈ [1, k], πs (zj ) = (yj , . . . , yj ). It follows from Lemma 5.29 that πs (P ) = R0l0 × · · · × Rsls . This means that P is dense in R.  Remark 5.32. We use below the following result which can be deduced easily from known results of ring theory: Let p be a prime number and n, k ∈ N. Then there exists s(n, k) ∈ N such that |R| ≤ s(n, k) for every k-generated commutative ring R of characteristic p satisfying the identity xn = x. Indeed, consider the free ring S with k free generators x1 , . . . , xk in the variety M defined by the identities xn = x, px = 0. The additive group of S is generated by elements xα1 1 , . . . , xαi i , where 1 ≤ i ≤ k and 0 ≤ αj ≤ n − 1 for j ∈ [1, i]. It follows that S is a vector Fp -space of finite dimension, say l. Set s(n, k) = pl . It follows that if F is a finite field, k ∈ N, then there exists n ∈ N such that the ring F n cannot be generated by k elements. Example 5.33. A semisimple vdW-ring with no finite number of topological generators: Fix p ∈ P, n ∈ N and consider the Galois field Fpn . Choose for every n ∈ N a number kn ∈ N such that the ring Rn = Fkpnn cannot be generated  by n elements. According to Theorem 2.8 the compact ring R = ∞ i=1 Rn is a vdW-ring. Obviously, R has no finite system of topological generators. A topological ring R (not necessarily with identity) is called indecomposable if for every decomposition R = A ⊕ B of R in a topological direct sum of ideals, A = R or A = 0. Recall [2](p. 99) that an abstract ring R with identity is called indecomposable provided 0, 1 are unique central idempotents of R. We notice that a topological ring (R, T )

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with identity is indecomposable if and only if it is indecomposable as an abstract ring. We will construct below examples of indecomposable vdW-rings which are not topologically finitely generated.

Example 5.34. Consider the ring R of a fixed prime characteristic p constructed in Example 5.33. Furnish M := R with a structure of a topological (R, R)-bimodule, where the left action of R on M is given by the multiplication of R and the right action of R on M is trivial: ∀r∈R ∀m∈M [mr = 0]. Consider the trivial extension S = R  M of R by M – see [32](p. 149). The ring S is compact and has a left identity (e, 0), where e is the identity of R. We claim that S is a vdW-ring. Since S has finite characteristic, we have to show that every cofinite ideal I of S is open. We have that I ∩ (R × {0}) = K × {0}, where K = {x ∈ R|(x, 0) ∈ I} is a cofinite ideal of R × {0}. The mapping f : R → R × {0}, r → (r, 0) is a ring isomorphism. Hence f −1 (K × {0}) = K is a cofinite ideal of R. Since ∀k∈K [(0, k) = (k, 0)(0, e)], we obtain that {0} × K ⊂ I(0, e) ⊂ I. This implies that K × K ⊂ I. Thus I is open. We notice that the center Z(S) of S is {0} : if (r, m) ∈ Z(S), then (r, m)(e, 0) = (e, 0)(r, m) ⇒ 0 = m. Furthermore, (r, 0)(0, e) = (0, e)(r, 0) ⇒ r = 0. We can embed S as an open ideal into a vdW-ring L with identity as it is indicated in Theorem 5.14. We note that S has no finite set of topological generators. Indeed, we observed in Example 5.33 that R has no finite system of topological generators. Since the mapping S → R, (r, m) → r is a surjective continuous ring homomorphism, the ring S also has no finite system of topological generators. The ring S is an open ideal of L. It follows from Remark 5.10 that L is not topologically finitely generated. We claim that L is an indecomposable ring. Indeed, let (α, s), α ∈ Fp , s ∈ S be a central idempotent of L. Then α = 1 or α = 0. For every s1 ∈ S, (0, s1 )(α, s) = (α, s)(0, s1 ) ⇒ αs1 + s1 s = αs1 + ss1 ⇒ s1 s = ss1 ⇒ s = 0.

Example 5.35. Fix a prime number p. Let kn ∈ N such that the minimal number of generators of the ring Fkpnn is ≥ n. Now we consider a sequence {Ln }n∈N of finite fields constructed as follows: L1 = L2 = · · · = Lk1 = Fp ; Lk1 +1 = Lk1 +2 = · · · = Lk2 = Fp2 ; · · · .

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Consider the ring ⎤ L1 0 0 · · · ⎢ L2 L2 0 · · · ⎥ ⎥ R=⎢ ⎣ L3 L 3 L 3 · · · ⎦ .. .. .. . . . ⎡

with the natural compact topology. We claim that R is a vdW-ring. By Lemma 2.10 it suffices to show that any cofinite ideal I of R is open. The subring ⎡

⎤ L1 0 0 · · · ⎢ 0 L2 0 · · · ⎥ ⎥ S=⎢ ⎣ 0 0 L3 · · · ⎦ .. .. .. . . . is compact and semisimple. According to the construction of S and by Theorem 2.8, S is a vdW-ring. Therefore there exists n ∈ N such that e = Σ∞ i=n Eii ∈ I, where Eii , i ≥ n is the matrix unit of R. The ideal H = {Σαst Est |s ≤ t, s ≥ n, αst ∈ Ls } is open in R and ∀h∈H [eh = h]. Hence H ⊂ I. This implies that I is open, and so R is a vdW-ring. The center of R consists of matrices of the form ⎡

⎤ α 0 0··· ⎢ 0 α 0··· ⎥ ⎢ ⎥ ⎣ 0 0 α··· ⎦, .. .. . . . . . where α ∈ Fp . It follows that ⎡

1 ⎢ 0 ⎢ ⎣ 0 .. .

⎤ 0 0··· 1 0··· ⎥ ⎥ 0 1··· ⎦ .. . . . .

is a unique central nonzero idempotent of R. Therefore, R is indecomposable. The mapping γ : R → S, sending each matrix from R to its diagonal, ∼ is a continuous surjective homomorphism. Since S =top n∈N Fkpnn , the ring S has no finite system of topological generators. Thus R has the same property.

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6. Rings of Type(AF) resp. Type(F) Lemma 6.1. Let R be a ring of type (AF). Then every cofinite subring R has this property. Proof. Assume the contrary. Let n be a natural number such that R has an infinite sequence I1 , I2 , · · · of ideals of index n. Let m = [R : R ]. Then every subring Ii has index mn in R. By Lemma 2.6 2 2 every Ii contains an ideal Ji of R of index ≤ mnm n +1 . It follows 2 2 that there exists k ∈ [1, mnm n +1 ] such that Js1 , Js2 , · · · have index k. Set Jsi = Ki . By the assumption the set {Ki |i ∈ N} is finite. Then K = ∩i∈N Ki is a cofinite ideal. This implies that the set {Jsi | ∈ N} is finite, a contradiction.  The following Lemma is obvious. Lemma 6.2. If a ring R is of type (AF), then every homomorphic image of R has this property. Lemma 6.3. Let R be a ring of finite characteristic which is of type (AF). Then for every n ∈ N, Rn is a cofinite ideal. Proof. By Lemma 6.2, R/Rn is a nilpotent ring of finite characteristic having property (AF). It suffices to prove the following: if R is a nilpotent ring of finite characteristic having property (AF), then R is finite. By Theorem 2.9, R/R2 is a direct sum of cyclic groups. This implies  that R2 is cofinite. By Lemma 3.13, R is finite. Theorem 6.4. Every cofinite ideal of a compact ring R is open if and only if R is of type (AF). Proof. ⇒: Assume the contrary, i.e., that there exists a sequence I1 , I2 , · · · of different ideals of index n for some n ∈ N. Then the quotient ring R/(∩∞ i=1 Ii ) is infinite, every cofinite ideal of it is open, satisfies the identity nx = 0 and has an infinite set of ideals of index n whose intersection is zero. Therefore, we may assume that R is a ring of finite characteristic and has a sequence I1 , I2 , · · · of different ideals of a given index n, ∩∞ i=1 Ii = {0} and each cofinite ideal of R is open. Since R is a subdirect product of rings of cardinality ≤ n, there exist natural numbers k and s with k > s such that ∀x∈R [xk = xs ] and J(R) is a nilpotent ring. The quotient ring R/J(R) is of type (AF) and satisfies the identities xk = xs , nx = 0. It follows that the cardinalities of all subrings of R/J(R) which are fields are ≤ k − s + 1. Moreover, if x is a nilpotent

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element of R/J(R), then xs = 0. It follows from [17](Theorem 16) that J(R) is cofinite. Then J(R) will be a nilpotent ring satisfying the identites xs = 0, nx = 0. According to Lemma 6.1, J(R) is of type (AF). Then Lemma 6.3 implies that J(R) is cofinite. By Lemma 3.13, J(R) is finite. So R is finite, a contradiction. ⇐: Assume the contrary, and let I be a nonclosed cofinite ideal. There exists n ∈ N such that nR ⊂ I. The ideal I/nR is cofinite and non-closed. Evidently, R/nR is a ring of typ (AF). We reduced the Theorem to the case, when the characteristic of R is n. The ideal I is open. By Lemma 6.1, I is a ring of type (AF). We reduced the proof to the case when R has characteristic n and R = I. The ring R/J(R) is of type (AF). By Theorem 2.8, R/J(R) is a vdW-ring. It follows that I + J(R) = R. The quotient ring R/I is radical and finite. Therefore, there exists m ∈ N such that Rm ⊂ I. By Lemma 6.3, Rm is cofinite, and by Lemma 3.16, Rm is open. Hence I is open, a contradiction.  Note 6.5. An analogous result for compact groups was proved by Smith and Wilson in [31]. Proposition 3.17 and Theorem 6.4 give Corollary 6.6. A compact ring R with identity (of finite characteristic) is a vdW-ring if and only if R is of type (AF). Theorem 6.4 implies Corollary 6.7. A profinite ring R is strongly complete if and only if R is of type (AF). The following result is obvious. Lemma 6.8. Let R be a ring of type (F). Then every continuous homomorphic image has this property. Remark 6.9. If in a ring R for every i ∈ N the set of open ideals of index i is finite, then for every i ∈ N the set of open subrings of index i is finite: (The proof is analogous with the proof of Lemma 6.1.) Assume the contrary: there exists n ∈ N such that R has an infinite set S1 , S2 , · · · of open subrings of index n. It follows from Lemma 2.6 2 that for every i ∈ N there exists an open ideal Pi ⊂ Si of index ≤ nn +1 . Denote P = ∩∞ i=1 Pi . The quotient ring R/P has the property that for every i the set of open ideals of index i is finite. It follows that the set S1 , S1 , · · · is finite, a contradiction.

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Remark 6.10. Let R be a compact ring having an open subring S of type (F). Then R has the same property: Assume on the contrary that R has an infinite set {I1 , I2 , · · · } of open ideals of a fixed index i. Then the set {S ∩ I1 , S ∩ I2 , · · · } is finite. Set P = ∩j∈N (S ∩ Ij ). Since there is a finite number of intermediate subrings between P and R, we obtain that the set {I1 , I2 , · · · } is finite, a contradiction. Theorem 6.11. Let R be a compact ring of type (F), and let J(R)2 = {0}. Then the ring R is of type (AF). Proof. We have to prove by Theorem 6.4 that each cofinite ideal I of R is open. Assume the contrary: let R be a ring satisfying the conditions of the theorem and having a nonopen cofinite ideal I. Then by Remark 6.10 I is a ring of type (F). Furthermore, J(I) = I ∩ J(R). Hence J(I)2 = {0}. We reduced the proof to the case when R = I. We may consider that R has a finite characteristic. Indeed, there exists n ∈ N such that nR ⊂ I. The ideal I/nR of R/nR is dense and non-open. By Lemma 6.8, R/nR is a ring of type (F). By Theorem 2.4 the quotient ring R/J(R) has identity e +J(R). According to Theorem 2.11 there exists an idempotent e ∈ e + J(R). The quotient ring R/J(R) is a ring of type (F). By Theorem 2.8, R/J(R) is a vdW-ring. Therefore, R/J(R) = (I +J(R))/J(R) and so R = I +J(R). We claim that e ∈ I. Indeed, there exist i ∈ I, j ∈ J(R) such that e = i + j. Then e = eie + eje. [21](Theorem (21.10)) implies J(eRe) = eRe ∩ J(R). The ring eRe is a ring with identity e. Since eje ∈ J(eRe), eie = e − eje is invertible in eRe by [21](Lemma (4.1)). This implies e ∈ I. Consider the continuous group homomorphism f : R → (1 − e)R(1 − e), r → (1 − e)r(1 − e). We note that (1 − e)R(1 − e) is a ring with trivial multiplication: if x, y ∈ R, then (1 − e)x(1 − e)(1 − e)y(1 − e) = (1 − e)x(1 − e)y(1 − e) = (x − ex)(y − ey)(1 − e) = (x − ex)(y − ey) − (x − ex)(y − ey)e = 0. We claim that f is a ring homomorphism. Indeed, ex−x ∈ J(R), ye− y ∈ J(R) for all x, y ∈ R. Then (ex − x)(ye − y) = 0 ⇒ exye − exy − xye + xy = 0 ⇒ xy = −exye + exy + xye ⇒ (1 − e)xy(1 − e) = 0 for all x, y ∈ R. This means that f is a continuous ring homomorphism. By Lemma 6.8, (1 − e)R(1 − e) is a ring of type (F). Since R has a finite characteristic, (1 − e)R(1 − e) is finite. Then f −1 (0) is open. If x ∈ f −1 (0), then (1 − e)x(1 − e) = 0. Hence x = ex + xe − exe ∈ I.  This means that f −1 (0) ⊂ I. Thus I is open, a contradiction.

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Lemma 6.12. Let R be a ring having only finitely many abstract subrings of index n for each integer n. Then every homomorphic image has this property. Lemma 6.13. Let R be a ring having only finitely many abstract subrings of index n for each integer n. Then every subring S of finite index inherits this property. Proof. Follows immediately from Lemma 2.6.



By [7](Remark 6.6) there are commutative strongly complete rings which are ”highly” not vdW-rings. We prove Theorem 6.14. For a profinite ring R the following conditions are equivalent: (i) R is strongly complete; (ii) R is a ring of type (AF); (iii) R has only countable many abstract subrings of finite index. If R has an identity, then these conditions are also equivalent to R being a vdW-ring. Proof. (i) ⇔ (ii) holds by Corollary 6.7. (ii) ⇒ (iii): Assume that there exists n such that R has an uncountable set of subrings of index n. It follows from Lemma 2.6 that R has 2 an uncountable set of ideals of some index k ≤ nn +1 , a contradiction. (iii) ⇒ (i): Assume that I is a nonclosed ideal of finite index n of R. Then I/nR is a nonclosed ideal of finite index of R/nR. Therefore, we can assume that the additive group of R is bounded. Further reduction: We can assume without loss of generality that I is dense. Indeed, I is an open ideal of R and, obviously, satisfies the condition (iii). Then there exists n ∈ N such that J(R)n ⊂ I. We have that R = I + J(R), because R/J(R) satisfies the condition (iii), is semisimple and I + J(R)/J(R) has a finite index and is open. Thus Rn ⊂ I. We claim that Rn is cofinite. Indeed, R/R2 is an Abelian bounded group satisfying the condition (iii), hence it is finite. Thus R/Rn is nilpotent with a bounded group and (R/Rn )/(R/Rn )2 = (R/Rn )/(R2 /Rn ) ∼ = R/R2 is finite, hence R/Rn is finite. By Lemma n 3.16, R is open, hence I is open. Thus I = R, a contradiction. The last assertion follows from Corollary 6.6.  Nikolov [25] and Segal [29] gave examples of profinite groups of type (F) which are not of type (AF). Problem 6.15. Is every ring of type (F) without zero-multiplication already of type (AF)?

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7. When are Compact Rings van der Waerden Rings? Lemma 7.1. Let R be a compact ring and A, B be two closed ideals which are finitely generated as left ideals. Then: (i) The product AB is a closed finitely generated left ideal; (ii) If additionally A, B are cofinite, then AB is cofinite. Proof. (i) Let A = Rx1 + · · · + Rxn + x1 , . . . , xn + and B = Ry1 + · · · + Rym +y1 , . . . , ym + , where n, m ∈ N. Denote C = Σ(i,j)∈[1,n]×[1,m] Rxi yj and D = xi yj |(i, j) ∈ [1, n] × [1, m]+ . If (i, j) ∈ [1, n] × [1, m] and r1 , r2 ∈ R, then r1 xi r2 yj ∈ (Rx1 + · · · + Rxn + x1 , . . . , xn + )yj ⊂ Rx1 yj + · · · + Rxn yj + x1 yj , . . . , xn yj + ⊂ C + D. If r ∈ R and i ∈ [1, n], j ∈ [1, m], then rxi yj ∈ C. If r ∈ R and i ∈ [1, n], j ∈ [1, m], then xi ryj ∈ (Rx1 + · · · + Rxn + x1 , . . . , xn + )yj ∈ Rx1 yj + · · · + Rxn yj + x1 yj , . . . , xn yj + ⊂ C + D. If i ∈ [1, n], j ∈ [1, m], then xi yj ∈ D. We have proved that AB = C + D. The left ideal Rx1 + · · · + Rxn is closed and the quotient group A/(Rx1 + · · · + Rxn ) is compact of cardinality ≤ ℵ0 . Therefore A/(Rx1 + · · · + Rxn ) is finite and so kA ⊂ Rx1 + · · · + Rxn for some k ∈ N. In particular, kxi yj ∈ (Rx1 + · · · + Rxn )yj ⊂ C for all ∈ [1, n] and j ∈ [1, m]. Therefore, kD ⊂ C and so AB/C is finite. Since C is compact, the ideal AB is compact. (ii) We have to show that AB is cofinite. Let R = K+A = L+B with finite L and K. Thus R = L+B = L+Ry1 +· · ·+Rym +y1 , . . . , ym + ⊂ L+ + (K + A)y1 + · · · + (K + A)ym + y1 , . . . , ym + ⊂ S+ + C, where S = L ∪ Ky1 ∪ · · · ∪ Kym ∪ {xi yj |i ∈ [1, n], j ∈ [1, m]} ∪ {y1 , . . . , ym }. Then R = S+ + C. Since R and C are compact, C is cofinite. It follows that C is open. Hence AB is open.  The next theorem is a generalization of [7](Corollary 4.6). Theorem 7.2. If R is a compact ring , R/J(R) is a vdW-ring and the Jacobson radical J = J(R) is a finitely generated left ideal, then R is a vdW-ring. Proof. Let H be a cofinite ideal. Then H + J(R)/J(R) is a cofinite ideal of R/J(R). Since R/J(R) is a vdW-ring, H +J(R)/J(R) is open. By Corollary 2.5 there exists a central idempotent e + J(R) of R/J(R) such that H + J(R)/J(R) = (R/J(R))(e + J(R)) = Re + J(R)/J(R).

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It follows that H + J(R) = Re + J(R). If J = Rx1 + · · · + Rxn + x1 , . . . , xn + , then H + J(R) = Re + Rx1 + · · · + Rxn + x1 , . . . , xn + = Re + Rx1 + · · · + Rxn + e, x1 , . . . , xn + . Since H is cofinite, there exists l ∈ N such that J(R)l ⊂ H. This implies (H + J(R))l ⊂ H. Hence (H + J(R))2l ⊂ H 2 . By Lemma 7.1, (H + J(R))2l is open. Thus H 2 is open. By Theorem 3.24, R is a vdW-ring.  Let R be a ring, and let B(R) be the set of all its central idempotents. It is known [12](Remarks at the end of Chapter 19) that B(R) becomes a Boolean ring if we define the addition a  b = a + b − 2ab for all a, b ∈ B(R) and a · b = ab for all a, b ∈ B(R). Indeed, (a + b − 2ab)2 = (a + b)2 − 4ab(a + b) + 4ab = a + 2ab + b − 4ab − 4ab + 4ab = a + b − 2ab, therefore a + b − 2ab ∈ B(R). Furthermore, if a, b, c ∈ B(R), then (a  b)  c = a  b + c − 2(a  b)c = a + b − 2ab + c − 2(a + b − 2ab)c a + b − 2ab − 2ac − 2bc + 4abc; a  (b  c) = a + b  c − 2a(b  c) = a + b + c − 2bc − 2a(b + c − 2bc) = a + b + c − 2bc − 2ab − 2ac + 4abc. If a ∈ B(R), then a  a = a + a − 2a2 = 0 and a · a = a. Furthermore, for every a, b, c ∈ B(R), a · (b  c) = a(b + c − 2bc) = ab + ac − 2abc; a·ba·c = ab  ac = ab + ac − 2abac = ab + ac − 2abc. Let now (R, T ) be a topological ring. Obviously B(R) is a closed subset and (B(R), , ·, T ) is a topological Boolean ring. Recall [20](Chapter iv, §10) that a nonzero element a of a Boolean ring R is called an atom provided there are no elements x = 0, a such that xa = x. A Boolean ring is called atomic provided for each x = 0 there exists at least one atom a such that ax = a. Remark 7.3. If R is ring and e is an atom of B(R), then Re is an indecomposable ring: Indeed, let g be a nonzero central idempotent of Re. Then for each x ∈ R, gx = gex = exg = xeg = xg, hence g ∈ B(R). Since ge = g, we obtain that g = e.

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We note that if e, e are two different atoms of a Boolean ring, then ee = 0. 

Theorem 7.4. Every compact ring (R, T ) with identity 1 is topologically isomorphic to a topological product of a family compact indecomposable rings. Proof. Let (B(R), , ·, T ) be the  Boolean compact ring constructed above. By Theorem 2.4, B(R) = α∈Ω Fα , where Fα = F2 and eα is the identity of Fα . Every idempotent eα is an atom of B(R). Hence Rα := Reα is an indecomposable ring. We have that 1 = Σα∈Ω eα (here Σ is the sum in the ring (B(R), , ·, T )). Since the idempotents eα are orthogonal,  1 = Σα∈Ω eα (here Σ is the sum in (R, T )). This implies that R = α∈Ω Rα .  Remark 7.5. Theorem 7.4 fails for compact rings without identity: and let A be the torsion subgroup of  Indeed,nlet p be a prime number, ∗ Z(p ). Then the group A of characters considered as a compact n∈N ring with trivial multiplication cannot be decomposed in a product of indecomposable rings.  Assume the contrary: Let A∗ = α∈Ω Aα , where each Aα is an indecomposable (topologically) group. Then A ∼ = ⊕α∈Ω A∗α . Obviously, ∗ each Aα is an indecomposable group. But by [13](Corollary 27.4) each indecomposable torsion group is cyclic. A contradiction, because by [19](Example 10.1.15) A cannot be decomposed into a direct sum of cyclic subgroups. Remark 7.6. Let R be a ring, I a quasiregular ideal and e an idempotent of R. If L is an ideal of R and e ∈ L + I, then e ∈ L: Indeed, eIe is an ideal of eRe. Since the Jacobson radical of eRe is eJ(R)e (see [16](Chapter III, §7, Proposition 1)) eIe is a quasiregular ideal of eRe. Let x ∈ L, y ∈ I, e = x + y. Then e = exe + eye. This implies e − eye = exe ∈ L. Since −eye is quasiregular in eRe, e − eye is inversable in eRe. It follows that e ∈ L. Theorem 7.7. Let {Rα }α∈Ω be a family of compact rings withidentity and for each α ∈ Ω let Iα be a quasiregular  ideal of Rα . Then α∈Ω Rα is a vdW-ring if and only if α∈Ω Rα / α∈Ω Iα is a vdW-ring and each Rα is a vdW-ring. Proof. ⇒: Obvious. of R ⇐: Denote for each α ∈ Ω by 1α the identity  α . Let I be a cofiideal (I + α∈Ω Iα )/ α∈Ω  Iα is cofinite. nite ideal of α∈ΩRα . The  Thus it is openin α∈Ω Rα / α∈Ω Iα . It follows that I + α∈Ω Iα is an open ideal of α∈Ω Rα . There exists a finite subset Ω0 of Ω such that

VAN DER WAERDEN RINGS





37



e = α∈Ω0 0α × β ∈Ω Iα . By Remark 7.6, e ∈ I. It folβ ∈ I+ / 0 1 α∈Ω R ⊂ I. This implies that lows immediately that α∈Ω0 {0α } × β ∈Ω   / 0 β   {0 })) + {0 . The ring I = (I ∩ ( α∈Ω0 Rα × β ∈Ω β α} × / α∈Ω β ∈Ω / 0 Rβ 0 0   R is a topological product of its ideals α∈Ω0 Rα × β ∈Ω α∈Ω / 0 {0β }     α R . The ring R × {0 } and α∈Ω0 {0α } × β ∈Ω β is topo/ 0 β α∈Ω0 α β ∈Ω / 0 logically isomorphic to a topological product of a finite number  of vdWrings. Therefore, it is a vdW-ring. This implies that I ∩ ( α∈Ω0 Rα ×    {0 }) is an open ideal of R × {0 }. Thus I is open β β β ∈Ω / 0 α∈Ω0 α β ∈Ω / 0  in α∈Ω Rα . Now apply Lemma 2.10. ringswith identity. Corollary  7.8. Let {Rα }α∈Ω be a family of compact  Then α∈Ω Rα is a vdW-ring if and only if α∈Ω Rα / α∈Ω J(Rα ) is a vdW-ring and each Rα is a vdW-ring. Corollary 7.8 and Theorem 7.4 reduce the study of vdW-rings with identity to the study of indecomposable vdW-rings. Lemma 7.9. If R is an Abelian compact ring with R2 = R, then R is a ring with identity. Proof. There exists an idempotent e ∈ R which is the identity modulo J(R). Thus R = Re ⊕ R(1 − e), a topological direct sum, and R(1 − e) is a radical ring. Since [R(1 − e)]2 = R(1 − e), it follows that R(1 − e) = {0}. Hence R = Re.  Abelian rings. Theorem  7.10. Let {Rα }α∈Ω be a family of compact   Then R = α∈Ω Rα is a vdW-ring if and only if α∈Ω Rα / α∈Ω J(Rα ) is a vdW-ring and almost all rings Rα are vdW-rings with identity. Proof. ⇒: We will show that almost all Rα are rings with identity. Since R and Rα are vdW-rings, Lemma 3.10 implies that R2 and Rα 2 2 are cofinite. Then  by Lemma 3.16, R2 = R2 and Rα2 for each  Rα = 2 2 2 2 2 α ∈ Ω. Now R = α∈Ω Rα . Thus R = R = α∈Ω Rα . Since R2 is cofinite, we obtain Rα = Rα2 for almost all α. By Lemma 7.9, Rα is a ring with identity for almost all α. ⇐: There exists a finite subset Ω0 of Ω such that Rα is a ring with identity for all α ∈ / Ω0 . Then use similar arguments as in the proof of Theorem 7.7.  Corollary 7.11. A compact commutative ring R is a vdW-ring if and only if R/J(R)2 is a vdW-ring.  Proof. ⇐: R is a product α∈Ω Rα × N , where Rα are local rings and  N is a radical ring. Since R/J(R)2 = ( α∈Ω Rα /J(Rα )2 ) × (N/N 2 ), Rα /J(Rα )2 (α ∈ Ω) and N/N 2 are vdW-rings. By Theorem 5.12 these

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rings are finite. By [32](Chapter 2, Theorem 6.43) Rα and N are rings with radical topology. By Lemma 5.2 they are vdW-rings. Since J(R)2 ⊂ J(R), the ring R/J(R) is a continuous homomorphic image of R/J(R)2 . Hence it is a vdW-ring. By Theorem 7.10, R is a vdW-ring. ⇒: Obvious.  Problem 7.12. Is it true that a compact ring R is a vdW-ring if and only if R/J(R)2 is a vdW-ring? Problem 7.13. (a) Classify those compact rings each closed subring of which is a vdW-ring. (b) Classify those compact rings each closed ideal of which is a vdWring. Acknowledgement With many thanks, the authors acknowledge an insightful, knowledgeable report from an anonymous referee. He helped us enhance both the scope of the paper and its style of presentation. References [1] M. P. Anderson, Subgroups of finite index in profinite groups, Pacific J. Math. 62(1976), 19–28 [2] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second Edition, 1992, Springer. [3] V. I. Arnautov and M. I. Ursul, On the uniqueness of topologies for some constructions of rings and modules, Siber. Math. J. 36(1995), 631–644 [in English]; Sibirsk. Mat. Zh. 36(1995), 735–751 [in Russian]. [4] G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Extensions of Rings and Modules, Birkh¨auser, 2013. [5] E. Cartan, Sur les repr´esentations lin´eaires des groupes clos, Comment. Math. Helv.2(1930), 269–283 [6] W. W. Comfort, S. Hern´andez, D. Remus and F. J. Trigos-Arrieta, Some open questions on topological groups, Research and Exposition in Mathematics 24(2000), 57–76. [7] W. W. Comfort, D. Remus and H. Szambien, Extending ring topologies, J. Algebra 232(2000), 21–47. [8] P. M. Cohn, Universal Algebra, Harper and Row. Publishers, 1965. [9] C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, New York, London, 1962. [10] E. K. van Douwen, The maximal totally bounded topology on G and the biggest minimal G-space for Abelian groups G, Topology Appl. 34(1990), 60–91. [11] C. Faith, Algebra I Rings, Modules and Categories, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, volume 190, SpringerVerlag 1973. [12] C. Faith, Algebra II Ring Theory, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen , volume 191, Springer Verlag, 1976.

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[13] L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York–San Francisco–London, 1970. [14] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, volume I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, volume 115, Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1963. [15] A. Hulanicki, Algebraic characterization of abelian divisible groups which admit compact topologies, Fundamenta Math. 44(1957), 192–197. [16] N. Jacobson, Structure of Rings, Amer. Math.Soc. Colloquium Publ., vol.37, Providence, RI, 1964(revised edition). [17] I. Kaplansky, Topological rings, Amer. J. Math. 69(1947), 153–183. [18] S. Kakutani, On cardinal numbers related with a compact group, Proc. Imperial Acad. Tokyo 19 (1943), 366–372. [19] M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer-Verlag, 1979. [20] K. Kuratowski and A. Mostowski, Set theory, North-Holland Publishing Company, 1967. [21] T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 1991. [22] S. Lang, Algebra, Springer, 2002. [23] J. Lewin, Subrings of finite index in finitely generated rings, J. Algebra 5(1967), 84–88. [24] A. I. Mal’cev, Algebraic Systems, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, volume 192, Springer Verlag, BerlinG¨ottingen-Heidelberg, 1973. [25] N. Nikolov, Algebraic properties of profinite groups, arXiv:1108.5130v6, 22 Feb 2012. [26] K. Numakura, Notes on compact rings with open radical, Czechoslovak Mathematical Journal 33(108)(1983), 101–106. [27] H. L. Peterson, Discontinuous characters and subgroups of finite index, Pacific J. Math. 44(1973), 683–691. [28] L. Ribes and P. Zaleskii, Profinite Groups, Second Edition, Springer-Verlag, 2010. [29] D. Segal, Remarks on profinite groups having few open subgroups, arXiv:1304.3893v3, 3 Apr 2017. [30] L. Small, An algebra seminar talk at USC on 10/18/1982. See: https://ncaglife.wordpress.com/2012/06/13/non-noetherian-subalgebrasof-finitely-generated-commutative-algebras/ [31] M. G. Smith and J. S. Wilson, On subgroups of finite index in compact Hausdorff groups, Archiv Math. 80(2003), 123–129. [32] M. Ursul, Topological Rings Satisfying Compactness Conditions, Kluwer Academic Publishers, 2002. [33] M. Ursul, Compact Rings and Their Generalizations, Kishinev, Stiintsa, 1991 [in Russian]. [34] M. Ursul, Strenthening the topologies of countably compact rings, Matematiceskie Issledovania 118(1990), 126–136 [in Russian]. [35] B. L. van der Waerden, Stetigkeitss¨ atze f¨ ur halbeinfache Liesche Gruppen, Math. Z. 36(1933),780–786. [36] S. Warner, Topological Rings, North-Holland Mathematics Studies 178, 1993.

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˘ [37] S. Warner, Compact rings and Stone-Cech compactifications, Arch. Math. 11(1960), 327–332. ¨ r Mathematik, Universita ¨ t Paderborn, Warburger Str. Institut fu 100, D-33095, Germany E-mail address: [email protected] Department of Mathematics and Computer Science, Lae, PNG University of Technology, Papua New Guinea E-mail address: [email protected]