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Algebraic properties of some factor rings of C(X)
Topology and its Applications 228 (2017) 79–91
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Topology and its Applications www.elsevier.com/locate/topol
Algebraic properties of some factor rings of C(X) H. Saberi, A. Taherifar ∗ , A.R. Olfati Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 18 August 2016 Received in revised form 22 May 2017 Accepted 22 May 2017 Available online 26 May 2017 MSC: primary 54C40 secondary 13A30 Keywords: Ring of continuous functions Maximal ideal Socle Essential ideal EIN -ring Goldie dimension Suslin number Isolated point EF -space Extremally disconnected space
a b s t r a c t In F. Azarpanah et al. (2008) [3] the authors have given some algebraic properties of the ring C(X)/CF (X), where CF (X) = OβX\I(X) . In this paper, first, we show that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite. Next, we generalize this work for rings C(X)/OA and C(X)/M A whenever A ⊆ βX (or just a closed one, in some cases) and then topological conditions on A for which every prime (maximal) ideal of C(X)/OA (resp., C(X)/M A ) is essential are characterized. We call a ring R an EIN -ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. It is shown for a closed subset A of βX that C(X)/OA is an EIN -ring if and only if C(X)/M A is an EIN -ring if and only if A is an EF -space. Minimal ideals, socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/M A ) are characterized. We prove also that dim(C(X)/OA ) ≥ dimC(X)/M A , where dimC(X)/M A denotes the Goldie dimension of C(X)/M A , and the inequality may be strict. © 2017 Published by Elsevier B.V.
1. Introduction All rings are assumed to be commutative and with unity. A non-zero ideal I of R is essential if it intersects non-trivially every ideal of R. For an ideal I of R, Ann(I) = {a ∈ R : aI = 0}. It is well known that in any ring, if Ann(I) = 0, then I is an essential ideal. And, in any reduced ring (i.e., ring with no non-zero nilpotent) an ideal I is essential if and only if Ann(I) = 0. The socle of R, Soc(R), is the intersection of all essential ideals of R which is also equal to the sum of all minimal ideals of R. In a similar fashion we use Socm (R) to denote the intersection of all essential maximal ideals of R. Let C(X) be the ring of real-valued continuous functions on an infinite completely regular Hausdorff space X and OA (resp., M A ) be the set
* Corresponding author. E-mail addresses: [email protected] (H. Saberi), [email protected], [email protected] (A. Taherifar), [email protected], [email protected] (A.R. Olfati). http://dx.doi.org/10.1016/j.topol.2017.05.009 0166-8641/© 2017 Published by Elsevier B.V.
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of all f ∈ C(X) such that A ⊆ intβX clβX Z(f ) (resp., A ⊆ clβX Z(f )), where A ⊆ βX. By definition OA ⊆ M A for each A ⊆ βX and they are two important ideals of C(X) (since by McKnight Theorem, for any ideal I of C(X), there exists a subset A of βX such that OA ⊆ I ⊆ M A , see [5]). The socle of C(X) is denoted by CF (X); it is characterized in [10] as the set of all functions which vanish everywhere except on a finite number of points of X. Also, Socm (C(X)) is characterized in [8, Theorem 2.2], as the set of all f for which Z(1 − f g) is finite for each g ∈ C(X). For more about essentiality and Soc, the reader is refereed to [1,3] and [6]. We recall that a collection Iα α∈I of nonzero ideals of a commutative ring R is said to be independent if Iβ ∩ Σβ=α∈I Iα = (0). The Goldie dimension of R is denoted by dim(R) and it is the smallest cardinal number a such that every independent set of nonzero ideals of R has cardinality less than or equal to a. We recall also that the smallest cardinal number b such that every family of pairwise disjoint nonempty open subsets of a space Y has cardinality less than or equal to b is called the cellularity or souslin number of Y and is denoted by c(Y ) or S(Y ). A commutative ring R is regular if for each a ∈ R, there exists a b ∈ R such that a = a2 b. It is well known that C(X) is a regular ring if and only if X is a P -space (i.e., a space X in which any Gδ -set is open). We denote by I, the ideal (I + OA )/OA (resp., (I + M A )/M A ) in C(X)/OA (resp., C(X)/M A ), where I is an ideal of C(X). Also, for f ∈ C(X), we mean of f ∈ C(X)/OA (resp., C(X)/M A ), is f + OA (resp., f + M A ). For a ring R, Max (R) is the space of maximal ideals of R, endowed with the Zariski (or hull-kernel) topology, for which the closed subsets are V(I) = {M ∈ Max R | I ⊆ M }}, for an ideal I of R. The intersection of all maximal ideals of R, Jacobson radical of the ring R, is denoted by J(R) which is equal to the set of all a ∈ R such that 1 − ab is a unit for all b ∈ R. For a subset I of a ring R, I R denote that I is an ideal of R. The properties of factor rings of a ring in Algebra are important and the factor ring may have some different properties from that ring. In [3], the authors have given some algebraic properties of C(X)/CF (X) as a factor rings of C(X), where CF (X) is the socle of C(X) and it was characterized as OβX\I(X) in [13]. As some famous ideals of C(X) are of the form OA or M A (e.g., CF (X), CK (X), Cψ (X) and Socm C(X)), it is important to know algebraic properties of the rings C(X)/OA and C(X)/M A , respectively. Our main aim in this article is to reveal important relations between C(X)/OA and C(X)/M A and apply these relations to get information about the space A. An outline of this article is as follows: After recalling some preliminary results in Section 1, which are frequently used in the subsequent sections, the next section deals with the factor rings of C(X) which are C-ring. It is proved that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite if and only if C(X)/CF (X) is embedded in a C-ring. As a general case, for an ideal I of C(X), we show that C(X)/I is a C-ring if and only if I = M A for some closed C-embedded subset A of υX. Section 3 is devoted to the characterization of essential ideals of C(X)/OA (resp., C(X)/M A ). First, we prove that C(X)/M A is a regular ring if and only if for each Z ∈ Z[X], clβX Z ∩ clβX A is clopen in clβX A. In order that C(X)/OA be a regular ring, it is necessary and sufficient that C(X)/M A be regular and A be a round subset of βX. The characterization of essential ideals of C(X)/OA (resp., C(X)/M A ) is the important part of this section. It is proved that an ideal I of C(X)/M A is essential if and only if intA θ(I) = ∅, where θ(I) = f ∈I clβX Z(f ). As a consequence, we prove that every prime ideal of C(X)/M A is essential if and only if every maximal ideal of C(X)/M A is essential if and only if A is a perfect subset of βX. We prove also that every maximal ideal of C(X)/OA is essential if and only if A does not contain a P -point of βX which is at same time an isolated point of A. We call a ring R an EIN -ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. We have shown that if A ⊆ βX is closed then, C(X)/OA is an EIN -ring if and only if C(X)/M A is an EIN -ring if and only if A is an EF -space. The final part of Section 3 shows that whenever A ⊆ βX is closed, then C(X)/M A is a Baer ring if and only if A is an extremally disconnected space. Finally, in Section 4, we give a characterization of Soc and Socm of C(X)/OA (resp., C(X)/M A ) whenever A is a closed subset of βX. It is proved that f ∈ Soc(C(X)/OA ) (resp., Soc(C(X)/M A )) if and only if A\(intβX clβX Z(f )∩A) is a finite subset of I(A)∩P (βX) (resp., a finite subset of I(A)), where I(A) is the set
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of isolated points of A and P (βX) is the set of all P -points of βX. Also, we prove that f ∈ Socm (C(X)/OA ) (resp., Socm (C(X)/M A )) if and only if for each g ∈ C(X)/OA (resp., C(X)/M A ), A ∩ clβX Z(1 − f g) is a finite subset of I(A) ∩ P (βX) (resp., a finite subset of I(A)). At the end of this section, we show that the Goldie dimension of C(X)/OA is not smaller than that of C(X)/M A which is equal to the Suslin number of A. 2. When is a factor ring of C(X) a C-ring? In this section we show that C(X)/CF (X) is a C-ring (i.e., a ring which is isomorphic to a ring of continuous functions) if and only if the set of isolated points of X is finite. For an ideal I of C(X), it is shown that C(X)/I is a C-ring if and only if I = M A for some C-embedded subset A of υX. We begin this section with the following lemma. Before, we recall that if π : X → Y is a continuous function, then the map π ˆ : C(Y ) → C(X), by π ˆ (f ) = f oπ is a ring homomorphism, where f ∈ C(Y ). Lemma 2.1 ([9, Theorem 10.6]). Every ring homomorphism C(X) → C(Y ), where X is a realcompact space, is induced by a continuous mapping Y → X. We use the above lemma to give the next result which is important in the sequel. Proposition 2.2. Let φ : C(X) → C(Y ) be a ring homomorphism. Then the kernel of φ is of the form M A , for some A ⊆ υX. Moreover every ideal of this form is the kernel of a ring homomorphism. Proof. By Lemma 2.1, there exists a continuous function π : Y → υX such that for every f ∈ C(X), φ(f ) = f υ oπ. If we put A = π(Y ), then it is obvious that for all f ∈ M A , φ(f ) = 0. Also if φ(f ) = 0 then f υ oπ = 0. This shows that f υ vanishes on π(Y ) and therefore f ∈ M A . Now for the last part, let A ⊆ υX. Define a map between C(X) and C(A) by φ(f ) = f υ |A . Then it is easy to see that φ is a ring homomorphism and ker(φ) = M A . 2 Corollary 2.3. Let I be a countably generated ideal in C(X). If C(X)/I is embedded in a ring of continuous functions, then Δ(I) = ∩f ∈I Z(f ) is open in X. Proof. Without loss of generality, we can assume that X is real compact. Assume {f1 , · · · , fn , · · · } be a countable subset of C(X) such that I = (f1 , · · · , fn , · · · ). If there exists a completely regular space Y such that C(X)/I is embedded in C(Y ), then by Proposition 2.2, there exits A ⊆ X such that I = 1 ∞ 1 2 (f1 , · · · , fn , · · · ) = MA . Define g = n=1 n |fn | 1 . For each n ∈ N, we have |fn | ≤ (2n (1 + |fn | 2 )g)2 . 2 (1+|fn | 2 )
Hence each fn is a multiple of g, by [9, 1D.3]. This implies that MA ⊆ g . Also for each n ∈ N, A ⊆ Z(fn ), so A ⊆ n∈N Z(fn ) = Z(g) and therefore MA = g . This shows that the principal ideal g is a z-ideal and hence Z(g) is open, i.e., Δ(I) = ∩f ∈I Z(f ) = Z(g) is open. 2 As a trivial consequence of the previous corollary and the fact that a principal ideal g is a z-ideal if and only if Z(g) is open (see [9]), we have the following result. Corollary 2.4. For f ∈ C(X), C(X)/(f ) is embedded in a ring of continuous functions if and only if Z(f ) is open in X. Whenever S(X) = ℵ0 , then it is shown by Azarpanah et al. that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite (see [3, Theorem 2.9]). Here, we present a generalization of this result for any space X.
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Proposition 2.5. The following statements are equivalent. (1) C(X)/CF (X) is embedded in a C-ring. (2) The set of isolated points of X is finite. (3) C(X)/CF (X) is a C-ring. Proof. Let (1) hold. We want to prove (2). By the fact that I(X) = I(υX), it is enough to prove that I(υX) is finite. We have C(X)/CF (X) ∼ = C(υX)/CF (υX). Suppose that C(X)/CF (X) is embedded in a ring of continuous functions say C(Y ). Then there exists a ring homomorphism φ : C(υX) → C(Y ) such that ker(φ) = CF (υX). By Proposition 2.2, there exists a subset A ⊆ υ(υX) = υ(X) such that CF (υX) = OβX\I(X) = MA . Hence A ⊆ υX \ I(X). If I(υX) is infinite, then it contains an infinite countable subset say B = {xn : n ∈ N}. We define the function f as follows. f (x) =
1 n,
0,
x = xn , n ∈ N x∈ / B.
It is easy to see that f is continuous on υX and f ∈ MA . But υX \ Z(f ) is infinite and so f ∈ / CF (υX). This is a contradiction and hence I(υX) must be finite. Now let (2) hold. It is obvious to see that if I(X) is finite, then C(X)/CF (X) ∼ = C(X \ I(X)). This shows that (3) holds. Trivially (3) implies (1). 2
Corollary 2.6. For an infinite discrete space X, of continuous functions.
x∈X Rx ⊕x∈X Rx ,
where each Rx = R is not embedded in any ring
Proposition 2.7. Let I be an ideal of C(X). Then C(X)/I is a C-ring if and only if there exists a closed C-embedded A ⊆ υX such that I = M A . Proof. Let Y be a completely regular space such that C(X)/I ∼ = C(Y ). Then there exists an homomorphism φ : C(X) → C(Y ) such that ker(φ) = I. By Lemma 2.1, there exists a continuous function π : Y → υX such that π ˆ = φ. By Proposition 2.2, we have ker(φ) = M π(Y ) . Now, by part 2 of Theorem 10.3 in [9], the subset π(Y ) is C-embedded in υX. On the other hand, it is easy to see that M A = M clυX A , where A = π(Y ) and A is C-embedded in υX, so clυX A is C-embedded in υX. Conversely, if a closed subset A ⊆ υX is C-embedded in υX, then we can define the map φ : C(X) → C(A) by φ(f ) = f υ |A . It is easy to see that φ is an onto homomorphism and ker(φ) = I = M A , i.e., C(X)/I is a C-ring. 2 We conclude this section with the following result which shows that whenever X = Σ, then C(X)/Socm C(X) is not a C-ring, (see [9, 4M]). Corollary 2.8. C(X)/Socm C(X) is a C-ring if and only if υX \ I(X) is dense in βX \ I(X) and C-embedded in υX. Proof. Suppose that C(X)/Socm C(X) is a C-ring. We have C(X)/Socm C(X) ∼ = C(υX)/Socm C(υX). Now by hypothesis and Proposition 2.7, there is a closed C-embedded subset A of υX such that Socm C(υX) = M βX\I(X) = M A . This equality implies that clβX A = βX \ I(X). Hence A = υX \ I(X). Conversely, it is obvious, by Proposition 2.7. 2 3. Algebraic properties of C(X)/O A (resp., C(X)/M A ) If we define φ from C(X)/OA to C(X)/M A , by φ(f + OA ) = f + M A , then it is easy to see that φ is a ring homomorphism and kerφ = M A /OA which is J(C(X)/OA ). Hence we have C(X)/M A ∼ =
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C(X)/OA /J(C(X)/OA ). This shows that C(X)/M A is a factor ring of C(X)/OA , as C(X)/OA is a factor ring of C(X). Since clβX A ∼ Max (C(X)/M A ) = Max (C(X)/OA ) and clβX Z(f ) ∼ V(f ), we observe that for f ∈ C(X) and a no-empty set A ⊆ βX, f ∈ C(X)/M A is a unit if and only if clβX A ∩clβX Z(f ) = ∅ and this is equivalent to the fact that f ∈ C(X)/OA is a unit. In a particular case, whenever A = βX \ I(X), then we have the following result. Corollary 3.1. The following are equivalent. (1) f ∈ C(X)/Socm (C(X)) is unit. (2) Z(f ) is a finite subset of I(X). (3) f ∈ C(X)/CF (X) is unit. Recall from [11], a subset A of βX is round if M A = OA . Theorem 3.2. Let A ⊆ βX. The following statements hold. (1) C(X)/M A is a regular ring if and only if for each Z ∈ Z[X], clβX Z ∩ clβX A is clopen in clβX A. (2) C(X)/OA is a regular ring if and only if A is round and C(X)/M A is a regular ring. Proof. (1) First, suppose that C(X)/M A is a regular ring and Z(f ) ∈ Z[X]. If clβX A ⊆ clβX Z(f ) or clβX A ∩clβX Z(f ) = ∅, then we are done. So we assume that clβX A clβX Z(f ) and clβX A ∩clβX Z(f ) = ∅. By hypothesis, there is g ∈ C(X)/M A , such that f = f 2 · g, i.e., f − f 2 g ∈ M A . Hence clβX A ⊆ clβX Z(f ) ∪ clβX Z(1 − f g). This shows that: clβX A = (clβX A ∩ clβX Z(f )) ∪ (clβX A ∩ clβX Z(1 − f g)), where clβX A ∩ clβX Z(f ) and clβX A ∩ clβX Z(1 − f g) are two disjoint non-empty closed sets in clβX A. Therefore clβX A ∩ clβX Z(f ) = clβX A \ clβX A ∩ clβX Z(1 − f g) is a clopen subset of clβX A. Conversely, let f ∈ C(X)/M A be a non-zero and non-unit element. Hence clβX Z(f ) ∩ clβX A = ∅. By hypothesis, clβX Z(f )∩clβX A is clopen in clβX A. Thus (clβX Z(f )∩clβX A) and clβX A\(clβX Z(f )∩clβX A) are disjoint closed subsets of clβX A and hence they are disjoint closed subsets in βX. So there is an h ∈ C ∗ (X) such 2 +1) that clβX A \ (clβX Z(f ) ∩ clβX A) ⊆ clβX Z(h) and clβX Z(f ) ∩ clβX Z(h) = ∅. Now we define g = fh(h2 +f 2 . Then g ∈ C(X) and Z(h) ⊆ Z(1 − f g). Thus clβX A ⊆ clβX Z(f ) ∪ clβX Z(1 − f g) = clβX Z(f (1 − f g)). This says that f (1 − f g) ∈ M A , i.e., f = f 2 · g. (2) Assume that C(X)/OA is a regular ring. First, we know that if R is a commutative regular ring, then J(R) = 0. Hence J(C(X)/OA ) = M A /OA = 0, i.e., A is round. On the other hand, we have C(X)/OA /J(C(X)/OA ) ∼ = C(X)/M A . Thus C(X)/OA ∼ = C(X)/M A , i.e., C(X)/M A is regular. Conversely, it is obvious. 2 The above theorem implies the following result. We recall that a point p ∈ βX is a P -point if Op = M p . We denote by P (βX), the set of all P -points of βX. Corollary 3.3. The following statements hold. (1) Let X be a compact space. Then C(X)/M A is a regular ring if and only if A is finite. (2) For p ∈ βX, C(X)/Op is a regular ring if and only if p is a P -point. We need the following result in the sequel which is easy to prove.
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Lemma 3.4. Let R be a ring such that J(R) = 0, and let I and J be ideals of R. Then the following statements hold. (1) f ∈ Ann(I) ⇐⇒ V(f ) ∪ V(I) = Max R. (2) V(Ann(I)) = V(Ann(J)) ⇐⇒ Ann(I) = Ann(J). (3) V(Ann(I)) = Max R \ intMax R V(I). Whenever we write Ann(I/M A ), this is the annihilator ideal of I/M A in C(X)/M A . Since for a closed subset A of βX, A = Max (C(X)/M A ) and for an ideal I of C(X), V(I) = θ(I), the following lemma is a particular case of Lemma 3.4. Lemma 3.5. Let A ⊆ βX be closed and I, J be ideals of C(X) containing M A . Then Ann(I/M A ) = Ann(J/M A ) if and only if intA (θ(I)) = intA (θ(J)). Corollary 3.6. Let I C(X) contain M A and A ⊆ βX be closed. Then I/M A is essential in C(X)/M A if and only if intA (θ(I)) = ∅. Proof. This is trivial by Lemma 3.5, and the fact that an ideal I in a reduced ring is essential if and only if Ann(I) = 0. 2 Proposition 3.7. Let A ⊆ βX be closed. The following are equivalent. (1) A is perfect with subspace topology. (2) Every prime ideal of C(X)/M A is essential. (3) Every maximal ideal of C(X)/M A is essential. Proof. We prove that (1) implies (2). First, let (1) hold and P/M A be a prime ideal of C(X)/M A . Then P ⊆ M x for an unique x ∈ A and we have intA θ(P ) = intA {x} = ∅, by hypothesis. Hence by Corollary 3.6, P/M A is essential. Trivially (2) implies (3). Now let (3) hold and x ∈ A. Then by hypothesis, M x /M A is an essential ideal of C(X)/M A and so by Corollary 3.6, intA θ(M x ) = intA {x} = ∅, i.e., x is not an isolated point in A. Thus A is a perfect space. 2 Corollary 3.8. The following statements are equivalent. (1) βX \ I(X) is perfect. (2) Every maximal ideal in C(X)/Socm (C(X)) is essential. (3) Every prime ideal in C(X)/Socm (C(X)) is essential. By Proposition 3.7, every prime ideal of C(X) is essential if and only if βX has no isolated points and this is equivalent to the fact that X has no isolated points. So we have the following result. Corollary 3.9. Every prime ideal of C(X) is essential if and only if X does not have any isolated point. We denote by P (βX) the set of all P -points of βX. Also, the set of isolated points of A ⊆ βX is denoted by I(A). Theorem 3.10. Let A ⊆ βX be closed. The following statements hold. (1) For x ∈ A, the maximal ideal M x /OA is essential in C(X)/OA if and only if x ∈ / I(A) ∩ P (βX).
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(2) Every maximal ideal of C(X)/OA is essential if and only if A does not contain a P -point which is at same time an isolated point of it. Proof. (1) First we assume that the maximal ideal M x /OA is essential and x ∈ I(A) ∩ P (βX). Then the ideal M x /OA = Ox /OA is essential. The point x is an isolated point in A, so there is a closed subset B of A such that {x} ∪ B = A. Thus the ideal OB /OA is non-zero and we have Ox /OA ∩ OB /OA = 0. This says that M x /OA is not essential, a contradiction. Conversely, suppose that M x /OA is not essential. Then there exists a non-zero ideal J/OA such that M x /OA ∩ J/OA = 0, i.e., M x ∩ J = OA . On the other hand, we have Ox /OA ∩ J/OA = 0. Thus Ox ∩ J = OA = M x ∩ J. This shows that M x ∩ J ⊆ Ox . Now let P be a minimal prime ideal containing Ox such that P J. Then P ⊇ M x , i.e., P = M x and hence M x = Ox . Also, the equality OA = M x ∩ J implies that A = clβX A = θ(OA ) = {x} ∪ θ(J) and x ∈ / θ(J) (if x ∈ θ(J), x A then J ⊆ M and hence O = J, a contradiction). Hence x is an isolated point of A which is at same time a P -point of βX. (2) This follows by (1). 2 Corollary 3.11. The space βX \ I(X) does not contain a P -point of βX which be an isolated point in it. Proof. It is a consequence of Theorem 3.10 and [7, Proposition 1.2].
2
We recall from [9], a point p ∈ βX is an F -point if the ideal Op is a prime ideal of C(X). Remark 3.12. We may have every maximal ideal in C(X)/OA is essential but it contains a prime ideal which is not essential. For this, let X = Σ = N ∪ {σ} which is [9, 4M]. Put A = {σ}. Then σ is not a P -point in βΣ (since Σ is not a P -space) and hence the only maximal ideal M σ /OA is essential in C(X)/OA , by Theorem 3.10 part (1). On the other hand, the point σ is an F -point in βΣ (for Σ is an F -space). Hence the ideal Oσ is prime in C(X) and so Oσ /OA = 0 is a prime ideal of C(X)/OA which is not essential. In the next result we characterize clopen subsets in A ⊆ βX by idempotents. Before we need the following lemma. Lemma 3.13. Let e be an idempotent of C(X)/OA . Then A ∩ intβX clβX Z(e) = A ∩ clβX Z(e). Proof. We have A ⊆ intβX (clβX Z(e) ∪ clβX Z(1 − e)). Hence there exists an open subset U such that A ⊆ U ⊆ clβX Z(e) ∪ clβX Z(1 − e). This implies that A ⊆ intβX clβX Z(e) ∪ clβX Z(1 − e) and similarly A ⊆ clβX Z(e) ∪ intβX clβX Z(1 − e). Thus A ⊆ intβX clβX Z(e) ∪ intβX clβX Z(1 − e), for clβX Z(e) ∩ clβX Z(1 − e) = ∅. Now, A ∩ clβX Z(e) ⊆ A \ A ∩ clβX Z(1 − e) ⊆ A ∩ intβX clβX Z(e). So we are done. 2 Lemma 3.14. Let A be a closed subset of βX. The following are equivalent. (1) H is a clopen subset of A. (2) H = intβX clβX Z(h) ∩ A for some idempotent h ∈ C(X)/OA . (3) H = clβX Z(f ) ∩ A for some idempotent f ∈ C(X)/M A . Proof. (1) ⇒ (2) Let H be a clopen subset in A. Then H and A \ H are two disjoint clopen subsets in A and hence they are two disjoint closed subsets in βX. Hence there are f, g ∈ C ∗ (X) such that H ⊆ 2 2 ) intβX clβX Z(f ) and A \ H ⊆ intβX clβX Z(g) and Z(f ) ∩ Z(g) = ∅. Now we define h = f f(1+g 2 +g 2 . Then h ∈ C(X), Z(g) ⊆ Z(1 − h) and Z(f ) ⊆ Z(h). Hence clβX Z(g) ⊆ clβX Z(1 − h) and clβX Z(f ) ⊆ clβX Z(h). By these we have A ⊆ intβX clβX Z(h) ∪ intβX clβX Z(1 − h) ⊆ intβX clβX Z(h(1 − h)), i.e., h ∈ C(X)/OA
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is an idempotent. On the other hand, H ⊆ A ∩ intβX clβX Z(h) and if p ∈ intβX clβX Z(h) ∩ A, then p ∈ A \ intβX clβX Z(1 − h) ∩ A and so p ∈ / A \ H, i.e., p ∈ H. Hence A ∩ intβX clβX Z(h) = H. (2) ⇒ (3) If h is an idempotent in C(X)/OA and H = intβX clβX Z(h) ∩ A, then by Lemma 3.13, H = clβX Z(h) ∩ A and trivially h is an idempotent of C(X)/M A . (3) ⇒ (1) H = A ∩ clβX Z(f ) = A \ clβX Z(1 − f ) ∩ A is clopen in A. For f is an idempotent in C(X)/M A . 2 The Lemma 3.14 implies that every clopen subset in βX is of the form clβX Z(f ) for some idempotent f ∈ C(X). Hence it is easily seen that X is connected if and only if βX is connected. Also, by Lemma 3.14, we obtain the following result. Corollary 3.15. For a closed subset A of βX, the following are equivalent. (1) C(X)/M A has no nontrivial idempotents. (2) The subspace A is connected. (3) C(X)/OA has no nontrivial idempotents. We recall from [14] that a space X is an EF -space if disjoint unions of clopens are completely separated. Definition 3.16. We call a ring R an EIN -ring if for each two orthogonal I, J R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. Any IN -ring (see [4]) is an EIN -ring. But if we put X as the sum of R (i.e., real number with usual topology) and N, then it is an EF -space which is not extremally disconnected, hence in this case C(X) is an EIN -ring which is not an IN -ring. In fact X is an EF -space if and only if C(X) is an EIN -ring (see the proof of the [14, Theorem 2.10]). Theorem 3.17. Let A ⊆ βX be closed. The following statements are equivalent. (1) C(X)/OA is an EIN -ring. (2) The space A is an EF -space. (3) C(X)/M A is an EIN -ring. Proof. First we prove that (1) and (2) are equivalent. Similarly, we can prove the equivalency of (2) and (3). Suppose that ( α∈S Aα ) ∩ ( β∈K Aβ ) = ∅, where for each α ∈ S, β ∈ K, Aα and Aβ are clopen subsets of A. Then by Lemma 3.14, for each α ∈ S, β ∈ K, there are idempotents eα and eβ in C(X)/OA such that Aα = A \ (intβX clβX Z(eα ) ∩ A) and Aβ = A \ (intβX clβX Z(eβ ) ∩ A). Hence (
A \ (intβX clβX Z(eα ) ∩ A)) ∩ (
α∈S
A \ (intβX clβX Z(eβ ) ∩ A)) = ∅.
β∈K
This implies that: A⊆(
α∈S
intβX clβX Z(eα )) ∪ (
intβX clβX Z(eβ )).(a)
α∈S
Now assume that I = eα : α ∈ S and J = eβ : β ∈ K . Then by (a), we have, IJ ⊆ OA , i.e., IJ = 0. By hypothesis, Ann(I) + Ann(J) = C(X)/OA . Hence there are f ∈ Ann(I) and g ∈ Ann(J) such that 1 = f + g, i.e., (1 − (f + g)) ∈ OA . Therefore for each α ∈ S, β ∈ K we have,
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A \ intβX clβX Z(eα ) ∩ A ⊆ clβX Z(f ), A \ intβX clβX Z(eβ ) ∩ A ⊆ clβX Z(g), and there is Z ∈ Z[OA ] such that Z ∩ Z(f + g) = ∅. These show that α∈S Aα ⊆ clβX Z(f ) ∩ A and Z(g) = ∅. Now by normality of A, these two disjoint β∈K Aβ ⊆ clβX Z(g) ∩ A and A ∩ clβX Z(f ) ∩ A ∩ clβX closed sets are completely separated in A and hence α∈S Aα and β∈K Aβ are completely separated. Next assume I, J C(X)/OA generated by two subsets {eα : α ∈ S} and {eβ : β ∈ K} of idempotents of C(X)/OA and IJ = 0. We must prove that Ann(I) + Ann(J) = C(X)/OA . But IJ = 0 implies that IJ ⊆ OA . This shows that A ⊆ θ(IJ) ⊆ θ(I) ∪ θ(J). Thus (A \ θ(I)) ∩ (A \ θ(J)) = A \ clβX Z(eα ) ∩ A) ∩ ( A \ clβX Z(eβ ) ∩ A) = ∅. ( α∈S
β∈K
So we have two disjoint unions of clopens in A, by Lemma 3.14. Since A is C ∗ -embedded in βX, the hypothesis implies that there are two disjoint zero-sets Z(f ) and Z(g) such that (
A \ clβX Z(eα ) ∩ A) ⊆ intβX clβX Z(f ),
α∈S
and (
A \ clβX Z(eβ ) ∩ A) ⊆ intβX clβX Z(g).
β∈K
Therefore A \ intβX clβX Z(f ) ⊆
clβX Z(eα ) ∩ A,
α∈S
and A \ intβX clβX Z(g) ⊆
clβX Z(eβ ) ∩ A.
β∈K 2
Therefore f h ∈ OA and gk ∈ OA for each h ∈ I and k ∈ J. Thus f +g 2 is a unit element in Ann(I) +Ann(J) (since A ∩ intβX clβX Z(f ) ∩ intβX clβX Z(g)) = ∅), i.e., Ann(I) + Ann(J) = C(X)/OA . 2 The above theorem implies the following result. Corollary 3.18. The following statements are equivalent. (1) C(X)/CF (X) is an EIN -ring. (2) βX \ I(X) is an EF -space. (3) C(X)/Socm C(X) is an EIN -ring. In the next result we give a topological characterization of C(X)/M A as a Baer ring. Proposition 3.19. For a closed subset A of βX, C(X)/M A is a Baer ring if and only if A is extremally disconnected.
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Proof. Let C(X)/M A be a Baer ring and F a closed subset of A. Then F is a closed subset of βX and so F = α∈S clβX Z(fα ), for some family {fα : α ∈ S}. Put I = fα : α ∈ S . Then there is an idempotent e ∈ C(X)/M A such that Ann(I) = Ann(e). Now Lemma 3.5 implies that intA F = intA ( α∈S clβX Z(fα ) ∩A) = intA (θ(I) ∩ A) = intA (clβX Z(e) ∩ A), where intA (clβX Z(e) ∩ A) is clopen in A. Now let A be extremally disconnected and I be an ideal of C(X)/M A . Then by hypothesis and Lemma 3.14, there is an idempotent f ∈ C(X)/M A such that intA θ(I) = clβX Z(f ) ∩ A = intA (clβX Z(f ) ∩ A). By Lemma 3.5, Ann(I/M A ) = Ann(f ). 2 4. Soc and Socm of C(X)/O A (resp., C(X)/M A ) In this section we give a characterization of socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/M A ). Essential ideals and the Goldie dimension of a ring R are two related concepts and we may have either dim(R) ≤ dim(R/I) or dim(R/I) ≤ dim(R), where I is an ideal of R. In this section, we prove that dim(C(X)/OA ) ≥ dim(C(X)/M A ) = S(A). We need the following lemma in the sequel. Lemma 4.1. Let A ⊆ βX be closed. The following statements hold. (1) Every minimal ideal of C(X)/OA is of the form OA\{x} , where x ∈ I(A) ∩ P (βX). (2) Every minimal ideal of C(X)/M A is of the form M A\{x} , where x ∈ I(A). Proof. (1) Let I/OA be a minimal ideal of C(X)/OA . Then there is an idempotent e ∈ C(X)/OA such that I/OA = e . Hence the ideal generated by 1 − e is maximal. So there is x ∈ A such that 1 − e = (1−e +OA )/OA = M x /OA . This equality shows that 1−e +OA = M x . Therefore θ(1−e +OA ) = θ(M x ), i.e., clβX Z(1 − e) ∩ A = {x}. But by Lemma 3.13, intβX clβX Z(1 − e) ∩ A = clβX Z(1 − e) ∩ A is clopen in A and hence x is an isolated point of A and we have (1 − e) ∈ Ox . This implies that M x = 1 − e + OA = Ox , i.e., x is a P -point in βX. Now we claim that e = OA\{x} /OA . To see this, let f ∈ OA\{x} /OA . Then A \intβX clβX Z(1 −e) ∩A = A \clβX Z(1 −e) ∩A = A \{x} ⊆ intβX clβX Z(f ). Thus A ⊆ intβX clβX Z(f ) ∪ intβX clβX Z(1 − e) ⊆ intβX clβX Z(f (1 − e)), i.e., f · (1 − e) = f · (1 − e) = 0. This says that f ∈ e . Hence OA\{x} /OA ⊆ e . We are to prove the other inclusion. As we have A ⊆ intβX clβX Z(e) ∪intβX clβX Z(1 −e). Thus A \ {x} = A \ intβX clβX Z(1 − e) ∩ A ⊆ intβX clβX Z(e). This shows that e ∈ OA\{x} and hence e ∈ OA\{x} /OA . So our claim is proved. Now let x ∈ I(A) be a P -point of βX. Then there is an idempotent e ∈ C(X)/OA such that intβX clβX Z(1 − e) ∩ A = clβX Z(1 − e) ∩ A = A \ A ∩ clβX Z(e) = {x}, by Lemma 3.14. This shows 1 − e = Ox /OA . Since x is a P -point, M x /OA = 1 − e . Hence the ideal e = OA\{x} /OA is minimal. This completes our proof. (2) The proof is similar to the (1). 2 The above lemma implies the following interesting result. Corollary 4.2. Every minimal ideal of C(X) is of the form M βX\{x} = OβX\{x} for some x ∈ I(X). Theorem 4.3. Let A ⊆ βX be closed. The following statements hold. (1) Soc(C(X)/OA ) = OA\I(A)∩P (βX) is equal to the set K = {f : A \ intβX clβX Z(f ) is a finite subset of I(A) ∩ P (βX)}. (2) Soc(C(X)/M A ) = OA\I(A) is equal to the set
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{f : A \ clβX Z(f ) is a finite subset of I(A)}. Proof. (1) Let f = f1 · e1 + f2 · e2 + ... + fn · en ∈ Soc(C(X)/OA ), where each ideal ei is minimal in C(X)/OA . Then by Lemma 4.1, for each 1 ≤ i ≤ n, there is xi ∈ I(A) ∩ P (βX) such that ei = OA\{xi } . So for each 1 ≤ i ≤ n, we have intβX clβX Z(ei ) ∩ A = A ∩ clβX Z(ei ) = A \ {xi }. Thus A \ intβX clβX Z(f ) ⊆ A \ (intβX clβX Z(f ) ∩ A) ⊆ {x1 , ..., xn } ⊆ I(A) ∩ P (βX). This shows that f ∈ OA\I(A)∩P (βX) . Also, it is easy to see that OA\I(A)∩P (βX) ⊆ K. Now suppose f ∈ C(X)/OA and A \ intβX clβX Z(f ) be a finite subset {x1 , ..., xn } of I(A) ∩ P (βX). For each 1 ≤ i ≤ n, there is an idempotent ei ∈ C(X)/OA such that intβX clβX Z(1 − ei ) ∩ A = A \ (intβX clβX Z(ei ) ∩A) = {xi }. This equality shows that 1−ei = Oxi /OA = Mix /OA . Hence by Lemma 4.1, n each ideal ei is minimal and we have A \ intβX clβX Z(f ) = i=1 intβX clβX Z(1 − ei ) ∩ A. This implies that A ⊆ intβX clβX Z(f (1 − e1 )(1 − e2 )...(1 − en )), i.e., f · (1 − e1 )... · (1 − en ) = 0. Therefore f ∈ e1 + ... + en ⊆ Soc(C(X)/OA ). (2) By [12, 2.3], f ∈ Soc(C(X)/M A ) if and only if Max (C(X)/M A ) \ V(f ) is finite and this is equivalent to the A \ clβX Z(f ) is finite. From Lemma 4.1(2), Soc(C(X)/M A ) ⊆ OA\I(A) and it is easy to prove that OA\I(A) ⊆ {f¯|A \ clβX Z(f ) is a finite subset of I(A) }. 2 Theorem 4.4. For a closed set A ⊆ βX, the following statements hold. (1) Socm (C(X)/M A ) = M A\I(A) is equal to the set S = {f : ∀g ∈ C(X)/M A , A ∩ clβX Z(1 − f g) is a finite subset of I(A)}. (2) Socm (C(X)/OA ) = M A\I(A)∩P (βX) is equal to the set {f : ∀g ∈ C(X)/OA , A ∩ clβX Z(1 − f g) is a finite subset of I(A) ∩ P (βX)}. Proof. (1) First, by Corollary 3.6, for each x ∈ A \ I(A) the ideal M x /M A is essential in C(X)/M A . Hence we have, Socm (C(X)/M A ) ⊆
M x /M A = M A\I(A) /M A .
x∈A\I(A)
Next, consider f ∈ M A\I(A) /M A . Then f ∈ M A\I(A) and hence for each g ∈ C(X)/M A , we have A ∩ clβX Z(1 − f g) ⊆ A \ clβX Z(f ) ⊆ I(A). As A ∩ clβX Z(1 − f g) is a compact subset of A, it must be a finite subset of I(A). This shows that M A\I(A) /M A ⊆ S. Now let f ∈ S and M p /M A be an essential maximal ideal such that f ∈ / M p /M A . By Corollary 3.6, p A p p ∈ / I(A). But f ∈ / M /M , implies f ∈ / M . So there is g ∈ C(X) such that 1 − f g ∈ M p . Therefore p ∈ clβX Z(1 − f g) ∩ A, i.e., p is an isolated point in A, a contradiction. (2) The proof is similar to the (1). 2
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Theorems 4.3 and 4.4 imply the following results. Corollary 4.5. For a closed subset A ⊆ βX, the following statements are equivalent. (1) Soc(C(X)/M A ) = 0. (2) Socm (C(X)/M A ) = 0. (3) A is a perfect subset of βX. The following result shows that we may have Soc(C(X)/OA ) = 0 but the ideal Socm (C(X)/OA ) = 0. For, it is enough to put A = βX \ I(X), then by Corollary 3.11, Soc(C(X)/OA ) = 0. However, A need not be a round subset of βX, see [2, Theorem 2.4]. Corollary 4.6. For a closed subset A ⊆ βX, the following statements hold. (1) Soc(C(X)/OA ) = 0 if and only if I(A) ∩ P (βX) = ∅. (2) Socm (C(X)/OA ) = 0 if and only if I(A) ∩ P (βX) = ∅ and A is a round subset of βX. We conclude this section with the following results. Theorem 4.7. Let A ⊆ βX. Then dim(C(X)/M A ) = S(A) ≤ dim(C(X)/OA ). Proof. By [12, Theorem 3.5], dim(C(X)/M A ) = S(Max (C(X)/M A )) = S(clβX A) = S(A). So it is enough to prove the last inequality. We can assume that A is closed. Now suppose that dimC(X)/OA = a and {Gα : α ∈ S} is a family of pairwise disjoint open subsets of A. For each α ∈ S, choose xα ∈ Gα and fα ∈ C(X) such that A \ Gα ⊆ intβX clβX Z(fα ) and xα ∈ / clβX Z(fα ). It is easy to see that fα is a nonzero element of C(X)/OA and for each two distinct α = β, fα ∩ fβ = (0). We claim the set {fα : α ∈ S} is an independent family of ideals in C(X)/OA . For, consider g ∈ fα ∩ β=α∈S fβ . Then n n g = fα · hα = i=1 rβi · fβi , where hα and each rβi are in C(X)/OA . Hence g · g = i=1 hα rβi fα fβi = 0. Thus g = 0. Therefore |S| ≤ a and hence S(A) ≤ a. 2 Corollary 4.8. The following statements hold. (1) dim(C(X)/M βX\X ) = S(βX \ X) ≤ dim(C(X)/CK (X)). (2) dim(C(X)/Socm C(X)) = S(βX \ I(X)) ≤ dim(C(X)/CF (X)). In [3, Remark 2.7], the authors have given an example to show that dim(C(X)/CF (X)) might be strictly greater than dimC(X). In the following, we obtain an easy proof for this result. Corollary 4.9. Let N be the set of positive integers with the discrete topology. Then dimC(N)dim(C(N)/ CF (N)). Proof. It is well known that CF (N) = M βN\N and S(βN \ N) = c = 2ℵ0 . By previous theorem, dim(C(N)/CF (N)) = c = 2ℵ0 . But dim(C(N)) = S(N) = ℵ0 . Hence the Goldie dimension of C(N) is strictly smaller than the Goldie dimension of C(N)/CF (N). 2 The following example shows that dim(C(X)/OA ) might be strictly greater than S(A). Example 4.10. Let X = N ∪ {ω} be the one point compactification of N. Then we have S(βX \ I(X)) = 1. By [9, 14F], Oω is not prime. Hence there are f, g ∈ C(X) such that f, g ∈ / Oω , but f g ∈ Oω . Thus f .g = 0
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and f , g are non-zero elements. This shows that the set {f , g } is an independent family of ideals of C(X)/Oω and hence dim(C(X)/Oω ) 1. Acknowledgement The authors would like to thank the referee for the careful reading of this paper and for pointing out some comments which led to a much improved paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F. Azarpanah, Essential ideals in C(X), Period. Math. Hung. 31 (1995) 105–112. F. Azarpanah, M. Ghirati, A. Taherifar, When is CF (X) = M βX\I(X) ?, Topol. Appl. 194 (2015) 22–25. F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 11 (2) (2008) 315–336. V. Camilo, W.K. Nicholson, M.F. Yousif, Ikeda–Nakayama rings, J. Algebra 226 (2000) 1001–1010. W.E. Dietrich, On the ideal structure of C(X), Trans. Am. Math. Soc. 152 (1970) 61–77. T. Dube, A note on the socle of certain types of f -rings, Bull. Iran. Math. Soc. 38 (2) (2012) 517–528. A.A. Estaji, O.A.S. Karamzadeh, On C(X) modulo its socle, Commun. Algebra 31 (2003) 1561–1571. M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. O.A.S. Karamzadeh, M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Am. Math. Soc. 93 (1) (1985) 179–184. M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. K. Samei, On the maximal spectrum of commutative semiprimitive rings, Colloq. Math. 83 (1) (2000) 5–13. A. Taherifar, Intersections of essential minimal prime ideals, Comment. Math. Univ. Carol. 55 (1) (2014) 121–130. A. Taherifar, Some new classes of topological spaces and annihilator ideals, Topol. Appl. 165 (2014) 84–97.
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Topology and its Applications www.elsevier.com/locate/topol
Algebraic properties of some factor rings of C(X) H. Saberi, A. Taherifar ∗ , A.R. Olfati Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 18 August 2016 Received in revised form 22 May 2017 Accepted 22 May 2017 Available online 26 May 2017 MSC: primary 54C40 secondary 13A30 Keywords: Ring of continuous functions Maximal ideal Socle Essential ideal EIN -ring Goldie dimension Suslin number Isolated point EF -space Extremally disconnected space
a b s t r a c t In F. Azarpanah et al. (2008) [3] the authors have given some algebraic properties of the ring C(X)/CF (X), where CF (X) = OβX\I(X) . In this paper, first, we show that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite. Next, we generalize this work for rings C(X)/OA and C(X)/M A whenever A ⊆ βX (or just a closed one, in some cases) and then topological conditions on A for which every prime (maximal) ideal of C(X)/OA (resp., C(X)/M A ) is essential are characterized. We call a ring R an EIN -ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. It is shown for a closed subset A of βX that C(X)/OA is an EIN -ring if and only if C(X)/M A is an EIN -ring if and only if A is an EF -space. Minimal ideals, socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/M A ) are characterized. We prove also that dim(C(X)/OA ) ≥ dimC(X)/M A , where dimC(X)/M A denotes the Goldie dimension of C(X)/M A , and the inequality may be strict. © 2017 Published by Elsevier B.V.
1. Introduction All rings are assumed to be commutative and with unity. A non-zero ideal I of R is essential if it intersects non-trivially every ideal of R. For an ideal I of R, Ann(I) = {a ∈ R : aI = 0}. It is well known that in any ring, if Ann(I) = 0, then I is an essential ideal. And, in any reduced ring (i.e., ring with no non-zero nilpotent) an ideal I is essential if and only if Ann(I) = 0. The socle of R, Soc(R), is the intersection of all essential ideals of R which is also equal to the sum of all minimal ideals of R. In a similar fashion we use Socm (R) to denote the intersection of all essential maximal ideals of R. Let C(X) be the ring of real-valued continuous functions on an infinite completely regular Hausdorff space X and OA (resp., M A ) be the set
* Corresponding author. E-mail addresses: [email protected] (H. Saberi), [email protected], [email protected] (A. Taherifar), [email protected], [email protected] (A.R. Olfati). http://dx.doi.org/10.1016/j.topol.2017.05.009 0166-8641/© 2017 Published by Elsevier B.V.
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of all f ∈ C(X) such that A ⊆ intβX clβX Z(f ) (resp., A ⊆ clβX Z(f )), where A ⊆ βX. By definition OA ⊆ M A for each A ⊆ βX and they are two important ideals of C(X) (since by McKnight Theorem, for any ideal I of C(X), there exists a subset A of βX such that OA ⊆ I ⊆ M A , see [5]). The socle of C(X) is denoted by CF (X); it is characterized in [10] as the set of all functions which vanish everywhere except on a finite number of points of X. Also, Socm (C(X)) is characterized in [8, Theorem 2.2], as the set of all f for which Z(1 − f g) is finite for each g ∈ C(X). For more about essentiality and Soc, the reader is refereed to [1,3] and [6]. We recall that a collection Iα α∈I of nonzero ideals of a commutative ring R is said to be independent if Iβ ∩ Σβ=α∈I Iα = (0). The Goldie dimension of R is denoted by dim(R) and it is the smallest cardinal number a such that every independent set of nonzero ideals of R has cardinality less than or equal to a. We recall also that the smallest cardinal number b such that every family of pairwise disjoint nonempty open subsets of a space Y has cardinality less than or equal to b is called the cellularity or souslin number of Y and is denoted by c(Y ) or S(Y ). A commutative ring R is regular if for each a ∈ R, there exists a b ∈ R such that a = a2 b. It is well known that C(X) is a regular ring if and only if X is a P -space (i.e., a space X in which any Gδ -set is open). We denote by I, the ideal (I + OA )/OA (resp., (I + M A )/M A ) in C(X)/OA (resp., C(X)/M A ), where I is an ideal of C(X). Also, for f ∈ C(X), we mean of f ∈ C(X)/OA (resp., C(X)/M A ), is f + OA (resp., f + M A ). For a ring R, Max (R) is the space of maximal ideals of R, endowed with the Zariski (or hull-kernel) topology, for which the closed subsets are V(I) = {M ∈ Max R | I ⊆ M }}, for an ideal I of R. The intersection of all maximal ideals of R, Jacobson radical of the ring R, is denoted by J(R) which is equal to the set of all a ∈ R such that 1 − ab is a unit for all b ∈ R. For a subset I of a ring R, I R denote that I is an ideal of R. The properties of factor rings of a ring in Algebra are important and the factor ring may have some different properties from that ring. In [3], the authors have given some algebraic properties of C(X)/CF (X) as a factor rings of C(X), where CF (X) is the socle of C(X) and it was characterized as OβX\I(X) in [13]. As some famous ideals of C(X) are of the form OA or M A (e.g., CF (X), CK (X), Cψ (X) and Socm C(X)), it is important to know algebraic properties of the rings C(X)/OA and C(X)/M A , respectively. Our main aim in this article is to reveal important relations between C(X)/OA and C(X)/M A and apply these relations to get information about the space A. An outline of this article is as follows: After recalling some preliminary results in Section 1, which are frequently used in the subsequent sections, the next section deals with the factor rings of C(X) which are C-ring. It is proved that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite if and only if C(X)/CF (X) is embedded in a C-ring. As a general case, for an ideal I of C(X), we show that C(X)/I is a C-ring if and only if I = M A for some closed C-embedded subset A of υX. Section 3 is devoted to the characterization of essential ideals of C(X)/OA (resp., C(X)/M A ). First, we prove that C(X)/M A is a regular ring if and only if for each Z ∈ Z[X], clβX Z ∩ clβX A is clopen in clβX A. In order that C(X)/OA be a regular ring, it is necessary and sufficient that C(X)/M A be regular and A be a round subset of βX. The characterization of essential ideals of C(X)/OA (resp., C(X)/M A ) is the important part of this section. It is proved that an ideal I of C(X)/M A is essential if and only if intA θ(I) = ∅, where θ(I) = f ∈I clβX Z(f ). As a consequence, we prove that every prime ideal of C(X)/M A is essential if and only if every maximal ideal of C(X)/M A is essential if and only if A is a perfect subset of βX. We prove also that every maximal ideal of C(X)/OA is essential if and only if A does not contain a P -point of βX which is at same time an isolated point of A. We call a ring R an EIN -ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. We have shown that if A ⊆ βX is closed then, C(X)/OA is an EIN -ring if and only if C(X)/M A is an EIN -ring if and only if A is an EF -space. The final part of Section 3 shows that whenever A ⊆ βX is closed, then C(X)/M A is a Baer ring if and only if A is an extremally disconnected space. Finally, in Section 4, we give a characterization of Soc and Socm of C(X)/OA (resp., C(X)/M A ) whenever A is a closed subset of βX. It is proved that f ∈ Soc(C(X)/OA ) (resp., Soc(C(X)/M A )) if and only if A\(intβX clβX Z(f )∩A) is a finite subset of I(A)∩P (βX) (resp., a finite subset of I(A)), where I(A) is the set
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of isolated points of A and P (βX) is the set of all P -points of βX. Also, we prove that f ∈ Socm (C(X)/OA ) (resp., Socm (C(X)/M A )) if and only if for each g ∈ C(X)/OA (resp., C(X)/M A ), A ∩ clβX Z(1 − f g) is a finite subset of I(A) ∩ P (βX) (resp., a finite subset of I(A)). At the end of this section, we show that the Goldie dimension of C(X)/OA is not smaller than that of C(X)/M A which is equal to the Suslin number of A. 2. When is a factor ring of C(X) a C-ring? In this section we show that C(X)/CF (X) is a C-ring (i.e., a ring which is isomorphic to a ring of continuous functions) if and only if the set of isolated points of X is finite. For an ideal I of C(X), it is shown that C(X)/I is a C-ring if and only if I = M A for some C-embedded subset A of υX. We begin this section with the following lemma. Before, we recall that if π : X → Y is a continuous function, then the map π ˆ : C(Y ) → C(X), by π ˆ (f ) = f oπ is a ring homomorphism, where f ∈ C(Y ). Lemma 2.1 ([9, Theorem 10.6]). Every ring homomorphism C(X) → C(Y ), where X is a realcompact space, is induced by a continuous mapping Y → X. We use the above lemma to give the next result which is important in the sequel. Proposition 2.2. Let φ : C(X) → C(Y ) be a ring homomorphism. Then the kernel of φ is of the form M A , for some A ⊆ υX. Moreover every ideal of this form is the kernel of a ring homomorphism. Proof. By Lemma 2.1, there exists a continuous function π : Y → υX such that for every f ∈ C(X), φ(f ) = f υ oπ. If we put A = π(Y ), then it is obvious that for all f ∈ M A , φ(f ) = 0. Also if φ(f ) = 0 then f υ oπ = 0. This shows that f υ vanishes on π(Y ) and therefore f ∈ M A . Now for the last part, let A ⊆ υX. Define a map between C(X) and C(A) by φ(f ) = f υ |A . Then it is easy to see that φ is a ring homomorphism and ker(φ) = M A . 2 Corollary 2.3. Let I be a countably generated ideal in C(X). If C(X)/I is embedded in a ring of continuous functions, then Δ(I) = ∩f ∈I Z(f ) is open in X. Proof. Without loss of generality, we can assume that X is real compact. Assume {f1 , · · · , fn , · · · } be a countable subset of C(X) such that I = (f1 , · · · , fn , · · · ). If there exists a completely regular space Y such that C(X)/I is embedded in C(Y ), then by Proposition 2.2, there exits A ⊆ X such that I = 1 ∞ 1 2 (f1 , · · · , fn , · · · ) = MA . Define g = n=1 n |fn | 1 . For each n ∈ N, we have |fn | ≤ (2n (1 + |fn | 2 )g)2 . 2 (1+|fn | 2 )
Hence each fn is a multiple of g, by [9, 1D.3]. This implies that MA ⊆ g . Also for each n ∈ N, A ⊆ Z(fn ), so A ⊆ n∈N Z(fn ) = Z(g) and therefore MA = g . This shows that the principal ideal g is a z-ideal and hence Z(g) is open, i.e., Δ(I) = ∩f ∈I Z(f ) = Z(g) is open. 2 As a trivial consequence of the previous corollary and the fact that a principal ideal g is a z-ideal if and only if Z(g) is open (see [9]), we have the following result. Corollary 2.4. For f ∈ C(X), C(X)/(f ) is embedded in a ring of continuous functions if and only if Z(f ) is open in X. Whenever S(X) = ℵ0 , then it is shown by Azarpanah et al. that C(X)/CF (X) is a C-ring if and only if the set of isolated points of X is finite (see [3, Theorem 2.9]). Here, we present a generalization of this result for any space X.
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Proposition 2.5. The following statements are equivalent. (1) C(X)/CF (X) is embedded in a C-ring. (2) The set of isolated points of X is finite. (3) C(X)/CF (X) is a C-ring. Proof. Let (1) hold. We want to prove (2). By the fact that I(X) = I(υX), it is enough to prove that I(υX) is finite. We have C(X)/CF (X) ∼ = C(υX)/CF (υX). Suppose that C(X)/CF (X) is embedded in a ring of continuous functions say C(Y ). Then there exists a ring homomorphism φ : C(υX) → C(Y ) such that ker(φ) = CF (υX). By Proposition 2.2, there exists a subset A ⊆ υ(υX) = υ(X) such that CF (υX) = OβX\I(X) = MA . Hence A ⊆ υX \ I(X). If I(υX) is infinite, then it contains an infinite countable subset say B = {xn : n ∈ N}. We define the function f as follows. f (x) =
1 n,
0,
x = xn , n ∈ N x∈ / B.
It is easy to see that f is continuous on υX and f ∈ MA . But υX \ Z(f ) is infinite and so f ∈ / CF (υX). This is a contradiction and hence I(υX) must be finite. Now let (2) hold. It is obvious to see that if I(X) is finite, then C(X)/CF (X) ∼ = C(X \ I(X)). This shows that (3) holds. Trivially (3) implies (1). 2
Corollary 2.6. For an infinite discrete space X, of continuous functions.
x∈X Rx ⊕x∈X Rx ,
where each Rx = R is not embedded in any ring
Proposition 2.7. Let I be an ideal of C(X). Then C(X)/I is a C-ring if and only if there exists a closed C-embedded A ⊆ υX such that I = M A . Proof. Let Y be a completely regular space such that C(X)/I ∼ = C(Y ). Then there exists an homomorphism φ : C(X) → C(Y ) such that ker(φ) = I. By Lemma 2.1, there exists a continuous function π : Y → υX such that π ˆ = φ. By Proposition 2.2, we have ker(φ) = M π(Y ) . Now, by part 2 of Theorem 10.3 in [9], the subset π(Y ) is C-embedded in υX. On the other hand, it is easy to see that M A = M clυX A , where A = π(Y ) and A is C-embedded in υX, so clυX A is C-embedded in υX. Conversely, if a closed subset A ⊆ υX is C-embedded in υX, then we can define the map φ : C(X) → C(A) by φ(f ) = f υ |A . It is easy to see that φ is an onto homomorphism and ker(φ) = I = M A , i.e., C(X)/I is a C-ring. 2 We conclude this section with the following result which shows that whenever X = Σ, then C(X)/Socm C(X) is not a C-ring, (see [9, 4M]). Corollary 2.8. C(X)/Socm C(X) is a C-ring if and only if υX \ I(X) is dense in βX \ I(X) and C-embedded in υX. Proof. Suppose that C(X)/Socm C(X) is a C-ring. We have C(X)/Socm C(X) ∼ = C(υX)/Socm C(υX). Now by hypothesis and Proposition 2.7, there is a closed C-embedded subset A of υX such that Socm C(υX) = M βX\I(X) = M A . This equality implies that clβX A = βX \ I(X). Hence A = υX \ I(X). Conversely, it is obvious, by Proposition 2.7. 2 3. Algebraic properties of C(X)/O A (resp., C(X)/M A ) If we define φ from C(X)/OA to C(X)/M A , by φ(f + OA ) = f + M A , then it is easy to see that φ is a ring homomorphism and kerφ = M A /OA which is J(C(X)/OA ). Hence we have C(X)/M A ∼ =
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C(X)/OA /J(C(X)/OA ). This shows that C(X)/M A is a factor ring of C(X)/OA , as C(X)/OA is a factor ring of C(X). Since clβX A ∼ Max (C(X)/M A ) = Max (C(X)/OA ) and clβX Z(f ) ∼ V(f ), we observe that for f ∈ C(X) and a no-empty set A ⊆ βX, f ∈ C(X)/M A is a unit if and only if clβX A ∩clβX Z(f ) = ∅ and this is equivalent to the fact that f ∈ C(X)/OA is a unit. In a particular case, whenever A = βX \ I(X), then we have the following result. Corollary 3.1. The following are equivalent. (1) f ∈ C(X)/Socm (C(X)) is unit. (2) Z(f ) is a finite subset of I(X). (3) f ∈ C(X)/CF (X) is unit. Recall from [11], a subset A of βX is round if M A = OA . Theorem 3.2. Let A ⊆ βX. The following statements hold. (1) C(X)/M A is a regular ring if and only if for each Z ∈ Z[X], clβX Z ∩ clβX A is clopen in clβX A. (2) C(X)/OA is a regular ring if and only if A is round and C(X)/M A is a regular ring. Proof. (1) First, suppose that C(X)/M A is a regular ring and Z(f ) ∈ Z[X]. If clβX A ⊆ clβX Z(f ) or clβX A ∩clβX Z(f ) = ∅, then we are done. So we assume that clβX A clβX Z(f ) and clβX A ∩clβX Z(f ) = ∅. By hypothesis, there is g ∈ C(X)/M A , such that f = f 2 · g, i.e., f − f 2 g ∈ M A . Hence clβX A ⊆ clβX Z(f ) ∪ clβX Z(1 − f g). This shows that: clβX A = (clβX A ∩ clβX Z(f )) ∪ (clβX A ∩ clβX Z(1 − f g)), where clβX A ∩ clβX Z(f ) and clβX A ∩ clβX Z(1 − f g) are two disjoint non-empty closed sets in clβX A. Therefore clβX A ∩ clβX Z(f ) = clβX A \ clβX A ∩ clβX Z(1 − f g) is a clopen subset of clβX A. Conversely, let f ∈ C(X)/M A be a non-zero and non-unit element. Hence clβX Z(f ) ∩ clβX A = ∅. By hypothesis, clβX Z(f )∩clβX A is clopen in clβX A. Thus (clβX Z(f )∩clβX A) and clβX A\(clβX Z(f )∩clβX A) are disjoint closed subsets of clβX A and hence they are disjoint closed subsets in βX. So there is an h ∈ C ∗ (X) such 2 +1) that clβX A \ (clβX Z(f ) ∩ clβX A) ⊆ clβX Z(h) and clβX Z(f ) ∩ clβX Z(h) = ∅. Now we define g = fh(h2 +f 2 . Then g ∈ C(X) and Z(h) ⊆ Z(1 − f g). Thus clβX A ⊆ clβX Z(f ) ∪ clβX Z(1 − f g) = clβX Z(f (1 − f g)). This says that f (1 − f g) ∈ M A , i.e., f = f 2 · g. (2) Assume that C(X)/OA is a regular ring. First, we know that if R is a commutative regular ring, then J(R) = 0. Hence J(C(X)/OA ) = M A /OA = 0, i.e., A is round. On the other hand, we have C(X)/OA /J(C(X)/OA ) ∼ = C(X)/M A . Thus C(X)/OA ∼ = C(X)/M A , i.e., C(X)/M A is regular. Conversely, it is obvious. 2 The above theorem implies the following result. We recall that a point p ∈ βX is a P -point if Op = M p . We denote by P (βX), the set of all P -points of βX. Corollary 3.3. The following statements hold. (1) Let X be a compact space. Then C(X)/M A is a regular ring if and only if A is finite. (2) For p ∈ βX, C(X)/Op is a regular ring if and only if p is a P -point. We need the following result in the sequel which is easy to prove.
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Lemma 3.4. Let R be a ring such that J(R) = 0, and let I and J be ideals of R. Then the following statements hold. (1) f ∈ Ann(I) ⇐⇒ V(f ) ∪ V(I) = Max R. (2) V(Ann(I)) = V(Ann(J)) ⇐⇒ Ann(I) = Ann(J). (3) V(Ann(I)) = Max R \ intMax R V(I). Whenever we write Ann(I/M A ), this is the annihilator ideal of I/M A in C(X)/M A . Since for a closed subset A of βX, A = Max (C(X)/M A ) and for an ideal I of C(X), V(I) = θ(I), the following lemma is a particular case of Lemma 3.4. Lemma 3.5. Let A ⊆ βX be closed and I, J be ideals of C(X) containing M A . Then Ann(I/M A ) = Ann(J/M A ) if and only if intA (θ(I)) = intA (θ(J)). Corollary 3.6. Let I C(X) contain M A and A ⊆ βX be closed. Then I/M A is essential in C(X)/M A if and only if intA (θ(I)) = ∅. Proof. This is trivial by Lemma 3.5, and the fact that an ideal I in a reduced ring is essential if and only if Ann(I) = 0. 2 Proposition 3.7. Let A ⊆ βX be closed. The following are equivalent. (1) A is perfect with subspace topology. (2) Every prime ideal of C(X)/M A is essential. (3) Every maximal ideal of C(X)/M A is essential. Proof. We prove that (1) implies (2). First, let (1) hold and P/M A be a prime ideal of C(X)/M A . Then P ⊆ M x for an unique x ∈ A and we have intA θ(P ) = intA {x} = ∅, by hypothesis. Hence by Corollary 3.6, P/M A is essential. Trivially (2) implies (3). Now let (3) hold and x ∈ A. Then by hypothesis, M x /M A is an essential ideal of C(X)/M A and so by Corollary 3.6, intA θ(M x ) = intA {x} = ∅, i.e., x is not an isolated point in A. Thus A is a perfect space. 2 Corollary 3.8. The following statements are equivalent. (1) βX \ I(X) is perfect. (2) Every maximal ideal in C(X)/Socm (C(X)) is essential. (3) Every prime ideal in C(X)/Socm (C(X)) is essential. By Proposition 3.7, every prime ideal of C(X) is essential if and only if βX has no isolated points and this is equivalent to the fact that X has no isolated points. So we have the following result. Corollary 3.9. Every prime ideal of C(X) is essential if and only if X does not have any isolated point. We denote by P (βX) the set of all P -points of βX. Also, the set of isolated points of A ⊆ βX is denoted by I(A). Theorem 3.10. Let A ⊆ βX be closed. The following statements hold. (1) For x ∈ A, the maximal ideal M x /OA is essential in C(X)/OA if and only if x ∈ / I(A) ∩ P (βX).
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(2) Every maximal ideal of C(X)/OA is essential if and only if A does not contain a P -point which is at same time an isolated point of it. Proof. (1) First we assume that the maximal ideal M x /OA is essential and x ∈ I(A) ∩ P (βX). Then the ideal M x /OA = Ox /OA is essential. The point x is an isolated point in A, so there is a closed subset B of A such that {x} ∪ B = A. Thus the ideal OB /OA is non-zero and we have Ox /OA ∩ OB /OA = 0. This says that M x /OA is not essential, a contradiction. Conversely, suppose that M x /OA is not essential. Then there exists a non-zero ideal J/OA such that M x /OA ∩ J/OA = 0, i.e., M x ∩ J = OA . On the other hand, we have Ox /OA ∩ J/OA = 0. Thus Ox ∩ J = OA = M x ∩ J. This shows that M x ∩ J ⊆ Ox . Now let P be a minimal prime ideal containing Ox such that P J. Then P ⊇ M x , i.e., P = M x and hence M x = Ox . Also, the equality OA = M x ∩ J implies that A = clβX A = θ(OA ) = {x} ∪ θ(J) and x ∈ / θ(J) (if x ∈ θ(J), x A then J ⊆ M and hence O = J, a contradiction). Hence x is an isolated point of A which is at same time a P -point of βX. (2) This follows by (1). 2 Corollary 3.11. The space βX \ I(X) does not contain a P -point of βX which be an isolated point in it. Proof. It is a consequence of Theorem 3.10 and [7, Proposition 1.2].
2
We recall from [9], a point p ∈ βX is an F -point if the ideal Op is a prime ideal of C(X). Remark 3.12. We may have every maximal ideal in C(X)/OA is essential but it contains a prime ideal which is not essential. For this, let X = Σ = N ∪ {σ} which is [9, 4M]. Put A = {σ}. Then σ is not a P -point in βΣ (since Σ is not a P -space) and hence the only maximal ideal M σ /OA is essential in C(X)/OA , by Theorem 3.10 part (1). On the other hand, the point σ is an F -point in βΣ (for Σ is an F -space). Hence the ideal Oσ is prime in C(X) and so Oσ /OA = 0 is a prime ideal of C(X)/OA which is not essential. In the next result we characterize clopen subsets in A ⊆ βX by idempotents. Before we need the following lemma. Lemma 3.13. Let e be an idempotent of C(X)/OA . Then A ∩ intβX clβX Z(e) = A ∩ clβX Z(e). Proof. We have A ⊆ intβX (clβX Z(e) ∪ clβX Z(1 − e)). Hence there exists an open subset U such that A ⊆ U ⊆ clβX Z(e) ∪ clβX Z(1 − e). This implies that A ⊆ intβX clβX Z(e) ∪ clβX Z(1 − e) and similarly A ⊆ clβX Z(e) ∪ intβX clβX Z(1 − e). Thus A ⊆ intβX clβX Z(e) ∪ intβX clβX Z(1 − e), for clβX Z(e) ∩ clβX Z(1 − e) = ∅. Now, A ∩ clβX Z(e) ⊆ A \ A ∩ clβX Z(1 − e) ⊆ A ∩ intβX clβX Z(e). So we are done. 2 Lemma 3.14. Let A be a closed subset of βX. The following are equivalent. (1) H is a clopen subset of A. (2) H = intβX clβX Z(h) ∩ A for some idempotent h ∈ C(X)/OA . (3) H = clβX Z(f ) ∩ A for some idempotent f ∈ C(X)/M A . Proof. (1) ⇒ (2) Let H be a clopen subset in A. Then H and A \ H are two disjoint clopen subsets in A and hence they are two disjoint closed subsets in βX. Hence there are f, g ∈ C ∗ (X) such that H ⊆ 2 2 ) intβX clβX Z(f ) and A \ H ⊆ intβX clβX Z(g) and Z(f ) ∩ Z(g) = ∅. Now we define h = f f(1+g 2 +g 2 . Then h ∈ C(X), Z(g) ⊆ Z(1 − h) and Z(f ) ⊆ Z(h). Hence clβX Z(g) ⊆ clβX Z(1 − h) and clβX Z(f ) ⊆ clβX Z(h). By these we have A ⊆ intβX clβX Z(h) ∪ intβX clβX Z(1 − h) ⊆ intβX clβX Z(h(1 − h)), i.e., h ∈ C(X)/OA
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is an idempotent. On the other hand, H ⊆ A ∩ intβX clβX Z(h) and if p ∈ intβX clβX Z(h) ∩ A, then p ∈ A \ intβX clβX Z(1 − h) ∩ A and so p ∈ / A \ H, i.e., p ∈ H. Hence A ∩ intβX clβX Z(h) = H. (2) ⇒ (3) If h is an idempotent in C(X)/OA and H = intβX clβX Z(h) ∩ A, then by Lemma 3.13, H = clβX Z(h) ∩ A and trivially h is an idempotent of C(X)/M A . (3) ⇒ (1) H = A ∩ clβX Z(f ) = A \ clβX Z(1 − f ) ∩ A is clopen in A. For f is an idempotent in C(X)/M A . 2 The Lemma 3.14 implies that every clopen subset in βX is of the form clβX Z(f ) for some idempotent f ∈ C(X). Hence it is easily seen that X is connected if and only if βX is connected. Also, by Lemma 3.14, we obtain the following result. Corollary 3.15. For a closed subset A of βX, the following are equivalent. (1) C(X)/M A has no nontrivial idempotents. (2) The subspace A is connected. (3) C(X)/OA has no nontrivial idempotents. We recall from [14] that a space X is an EF -space if disjoint unions of clopens are completely separated. Definition 3.16. We call a ring R an EIN -ring if for each two orthogonal I, J R which are generated by two subsets of idempotents, Ann(I) + Ann(J) = R. Any IN -ring (see [4]) is an EIN -ring. But if we put X as the sum of R (i.e., real number with usual topology) and N, then it is an EF -space which is not extremally disconnected, hence in this case C(X) is an EIN -ring which is not an IN -ring. In fact X is an EF -space if and only if C(X) is an EIN -ring (see the proof of the [14, Theorem 2.10]). Theorem 3.17. Let A ⊆ βX be closed. The following statements are equivalent. (1) C(X)/OA is an EIN -ring. (2) The space A is an EF -space. (3) C(X)/M A is an EIN -ring. Proof. First we prove that (1) and (2) are equivalent. Similarly, we can prove the equivalency of (2) and (3). Suppose that ( α∈S Aα ) ∩ ( β∈K Aβ ) = ∅, where for each α ∈ S, β ∈ K, Aα and Aβ are clopen subsets of A. Then by Lemma 3.14, for each α ∈ S, β ∈ K, there are idempotents eα and eβ in C(X)/OA such that Aα = A \ (intβX clβX Z(eα ) ∩ A) and Aβ = A \ (intβX clβX Z(eβ ) ∩ A). Hence (
A \ (intβX clβX Z(eα ) ∩ A)) ∩ (
α∈S
A \ (intβX clβX Z(eβ ) ∩ A)) = ∅.
β∈K
This implies that: A⊆(
α∈S
intβX clβX Z(eα )) ∪ (
intβX clβX Z(eβ )).(a)
α∈S
Now assume that I = eα : α ∈ S and J = eβ : β ∈ K . Then by (a), we have, IJ ⊆ OA , i.e., IJ = 0. By hypothesis, Ann(I) + Ann(J) = C(X)/OA . Hence there are f ∈ Ann(I) and g ∈ Ann(J) such that 1 = f + g, i.e., (1 − (f + g)) ∈ OA . Therefore for each α ∈ S, β ∈ K we have,
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A \ intβX clβX Z(eα ) ∩ A ⊆ clβX Z(f ), A \ intβX clβX Z(eβ ) ∩ A ⊆ clβX Z(g), and there is Z ∈ Z[OA ] such that Z ∩ Z(f + g) = ∅. These show that α∈S Aα ⊆ clβX Z(f ) ∩ A and Z(g) = ∅. Now by normality of A, these two disjoint β∈K Aβ ⊆ clβX Z(g) ∩ A and A ∩ clβX Z(f ) ∩ A ∩ clβX closed sets are completely separated in A and hence α∈S Aα and β∈K Aβ are completely separated. Next assume I, J C(X)/OA generated by two subsets {eα : α ∈ S} and {eβ : β ∈ K} of idempotents of C(X)/OA and IJ = 0. We must prove that Ann(I) + Ann(J) = C(X)/OA . But IJ = 0 implies that IJ ⊆ OA . This shows that A ⊆ θ(IJ) ⊆ θ(I) ∪ θ(J). Thus (A \ θ(I)) ∩ (A \ θ(J)) = A \ clβX Z(eα ) ∩ A) ∩ ( A \ clβX Z(eβ ) ∩ A) = ∅. ( α∈S
β∈K
So we have two disjoint unions of clopens in A, by Lemma 3.14. Since A is C ∗ -embedded in βX, the hypothesis implies that there are two disjoint zero-sets Z(f ) and Z(g) such that (
A \ clβX Z(eα ) ∩ A) ⊆ intβX clβX Z(f ),
α∈S
and (
A \ clβX Z(eβ ) ∩ A) ⊆ intβX clβX Z(g).
β∈K
Therefore A \ intβX clβX Z(f ) ⊆
clβX Z(eα ) ∩ A,
α∈S
and A \ intβX clβX Z(g) ⊆
clβX Z(eβ ) ∩ A.
β∈K 2
Therefore f h ∈ OA and gk ∈ OA for each h ∈ I and k ∈ J. Thus f +g 2 is a unit element in Ann(I) +Ann(J) (since A ∩ intβX clβX Z(f ) ∩ intβX clβX Z(g)) = ∅), i.e., Ann(I) + Ann(J) = C(X)/OA . 2 The above theorem implies the following result. Corollary 3.18. The following statements are equivalent. (1) C(X)/CF (X) is an EIN -ring. (2) βX \ I(X) is an EF -space. (3) C(X)/Socm C(X) is an EIN -ring. In the next result we give a topological characterization of C(X)/M A as a Baer ring. Proposition 3.19. For a closed subset A of βX, C(X)/M A is a Baer ring if and only if A is extremally disconnected.
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Proof. Let C(X)/M A be a Baer ring and F a closed subset of A. Then F is a closed subset of βX and so F = α∈S clβX Z(fα ), for some family {fα : α ∈ S}. Put I = fα : α ∈ S . Then there is an idempotent e ∈ C(X)/M A such that Ann(I) = Ann(e). Now Lemma 3.5 implies that intA F = intA ( α∈S clβX Z(fα ) ∩A) = intA (θ(I) ∩ A) = intA (clβX Z(e) ∩ A), where intA (clβX Z(e) ∩ A) is clopen in A. Now let A be extremally disconnected and I be an ideal of C(X)/M A . Then by hypothesis and Lemma 3.14, there is an idempotent f ∈ C(X)/M A such that intA θ(I) = clβX Z(f ) ∩ A = intA (clβX Z(f ) ∩ A). By Lemma 3.5, Ann(I/M A ) = Ann(f ). 2 4. Soc and Socm of C(X)/O A (resp., C(X)/M A ) In this section we give a characterization of socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/M A ). Essential ideals and the Goldie dimension of a ring R are two related concepts and we may have either dim(R) ≤ dim(R/I) or dim(R/I) ≤ dim(R), where I is an ideal of R. In this section, we prove that dim(C(X)/OA ) ≥ dim(C(X)/M A ) = S(A). We need the following lemma in the sequel. Lemma 4.1. Let A ⊆ βX be closed. The following statements hold. (1) Every minimal ideal of C(X)/OA is of the form OA\{x} , where x ∈ I(A) ∩ P (βX). (2) Every minimal ideal of C(X)/M A is of the form M A\{x} , where x ∈ I(A). Proof. (1) Let I/OA be a minimal ideal of C(X)/OA . Then there is an idempotent e ∈ C(X)/OA such that I/OA = e . Hence the ideal generated by 1 − e is maximal. So there is x ∈ A such that 1 − e = (1−e +OA )/OA = M x /OA . This equality shows that 1−e +OA = M x . Therefore θ(1−e +OA ) = θ(M x ), i.e., clβX Z(1 − e) ∩ A = {x}. But by Lemma 3.13, intβX clβX Z(1 − e) ∩ A = clβX Z(1 − e) ∩ A is clopen in A and hence x is an isolated point of A and we have (1 − e) ∈ Ox . This implies that M x = 1 − e + OA = Ox , i.e., x is a P -point in βX. Now we claim that e = OA\{x} /OA . To see this, let f ∈ OA\{x} /OA . Then A \intβX clβX Z(1 −e) ∩A = A \clβX Z(1 −e) ∩A = A \{x} ⊆ intβX clβX Z(f ). Thus A ⊆ intβX clβX Z(f ) ∪ intβX clβX Z(1 − e) ⊆ intβX clβX Z(f (1 − e)), i.e., f · (1 − e) = f · (1 − e) = 0. This says that f ∈ e . Hence OA\{x} /OA ⊆ e . We are to prove the other inclusion. As we have A ⊆ intβX clβX Z(e) ∪intβX clβX Z(1 −e). Thus A \ {x} = A \ intβX clβX Z(1 − e) ∩ A ⊆ intβX clβX Z(e). This shows that e ∈ OA\{x} and hence e ∈ OA\{x} /OA . So our claim is proved. Now let x ∈ I(A) be a P -point of βX. Then there is an idempotent e ∈ C(X)/OA such that intβX clβX Z(1 − e) ∩ A = clβX Z(1 − e) ∩ A = A \ A ∩ clβX Z(e) = {x}, by Lemma 3.14. This shows 1 − e = Ox /OA . Since x is a P -point, M x /OA = 1 − e . Hence the ideal e = OA\{x} /OA is minimal. This completes our proof. (2) The proof is similar to the (1). 2 The above lemma implies the following interesting result. Corollary 4.2. Every minimal ideal of C(X) is of the form M βX\{x} = OβX\{x} for some x ∈ I(X). Theorem 4.3. Let A ⊆ βX be closed. The following statements hold. (1) Soc(C(X)/OA ) = OA\I(A)∩P (βX) is equal to the set K = {f : A \ intβX clβX Z(f ) is a finite subset of I(A) ∩ P (βX)}. (2) Soc(C(X)/M A ) = OA\I(A) is equal to the set
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{f : A \ clβX Z(f ) is a finite subset of I(A)}. Proof. (1) Let f = f1 · e1 + f2 · e2 + ... + fn · en ∈ Soc(C(X)/OA ), where each ideal ei is minimal in C(X)/OA . Then by Lemma 4.1, for each 1 ≤ i ≤ n, there is xi ∈ I(A) ∩ P (βX) such that ei = OA\{xi } . So for each 1 ≤ i ≤ n, we have intβX clβX Z(ei ) ∩ A = A ∩ clβX Z(ei ) = A \ {xi }. Thus A \ intβX clβX Z(f ) ⊆ A \ (intβX clβX Z(f ) ∩ A) ⊆ {x1 , ..., xn } ⊆ I(A) ∩ P (βX). This shows that f ∈ OA\I(A)∩P (βX) . Also, it is easy to see that OA\I(A)∩P (βX) ⊆ K. Now suppose f ∈ C(X)/OA and A \ intβX clβX Z(f ) be a finite subset {x1 , ..., xn } of I(A) ∩ P (βX). For each 1 ≤ i ≤ n, there is an idempotent ei ∈ C(X)/OA such that intβX clβX Z(1 − ei ) ∩ A = A \ (intβX clβX Z(ei ) ∩A) = {xi }. This equality shows that 1−ei = Oxi /OA = Mix /OA . Hence by Lemma 4.1, n each ideal ei is minimal and we have A \ intβX clβX Z(f ) = i=1 intβX clβX Z(1 − ei ) ∩ A. This implies that A ⊆ intβX clβX Z(f (1 − e1 )(1 − e2 )...(1 − en )), i.e., f · (1 − e1 )... · (1 − en ) = 0. Therefore f ∈ e1 + ... + en ⊆ Soc(C(X)/OA ). (2) By [12, 2.3], f ∈ Soc(C(X)/M A ) if and only if Max (C(X)/M A ) \ V(f ) is finite and this is equivalent to the A \ clβX Z(f ) is finite. From Lemma 4.1(2), Soc(C(X)/M A ) ⊆ OA\I(A) and it is easy to prove that OA\I(A) ⊆ {f¯|A \ clβX Z(f ) is a finite subset of I(A) }. 2 Theorem 4.4. For a closed set A ⊆ βX, the following statements hold. (1) Socm (C(X)/M A ) = M A\I(A) is equal to the set S = {f : ∀g ∈ C(X)/M A , A ∩ clβX Z(1 − f g) is a finite subset of I(A)}. (2) Socm (C(X)/OA ) = M A\I(A)∩P (βX) is equal to the set {f : ∀g ∈ C(X)/OA , A ∩ clβX Z(1 − f g) is a finite subset of I(A) ∩ P (βX)}. Proof. (1) First, by Corollary 3.6, for each x ∈ A \ I(A) the ideal M x /M A is essential in C(X)/M A . Hence we have, Socm (C(X)/M A ) ⊆
M x /M A = M A\I(A) /M A .
x∈A\I(A)
Next, consider f ∈ M A\I(A) /M A . Then f ∈ M A\I(A) and hence for each g ∈ C(X)/M A , we have A ∩ clβX Z(1 − f g) ⊆ A \ clβX Z(f ) ⊆ I(A). As A ∩ clβX Z(1 − f g) is a compact subset of A, it must be a finite subset of I(A). This shows that M A\I(A) /M A ⊆ S. Now let f ∈ S and M p /M A be an essential maximal ideal such that f ∈ / M p /M A . By Corollary 3.6, p A p p ∈ / I(A). But f ∈ / M /M , implies f ∈ / M . So there is g ∈ C(X) such that 1 − f g ∈ M p . Therefore p ∈ clβX Z(1 − f g) ∩ A, i.e., p is an isolated point in A, a contradiction. (2) The proof is similar to the (1). 2
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Theorems 4.3 and 4.4 imply the following results. Corollary 4.5. For a closed subset A ⊆ βX, the following statements are equivalent. (1) Soc(C(X)/M A ) = 0. (2) Socm (C(X)/M A ) = 0. (3) A is a perfect subset of βX. The following result shows that we may have Soc(C(X)/OA ) = 0 but the ideal Socm (C(X)/OA ) = 0. For, it is enough to put A = βX \ I(X), then by Corollary 3.11, Soc(C(X)/OA ) = 0. However, A need not be a round subset of βX, see [2, Theorem 2.4]. Corollary 4.6. For a closed subset A ⊆ βX, the following statements hold. (1) Soc(C(X)/OA ) = 0 if and only if I(A) ∩ P (βX) = ∅. (2) Socm (C(X)/OA ) = 0 if and only if I(A) ∩ P (βX) = ∅ and A is a round subset of βX. We conclude this section with the following results. Theorem 4.7. Let A ⊆ βX. Then dim(C(X)/M A ) = S(A) ≤ dim(C(X)/OA ). Proof. By [12, Theorem 3.5], dim(C(X)/M A ) = S(Max (C(X)/M A )) = S(clβX A) = S(A). So it is enough to prove the last inequality. We can assume that A is closed. Now suppose that dimC(X)/OA = a and {Gα : α ∈ S} is a family of pairwise disjoint open subsets of A. For each α ∈ S, choose xα ∈ Gα and fα ∈ C(X) such that A \ Gα ⊆ intβX clβX Z(fα ) and xα ∈ / clβX Z(fα ). It is easy to see that fα is a nonzero element of C(X)/OA and for each two distinct α = β, fα ∩ fβ = (0). We claim the set {fα : α ∈ S} is an independent family of ideals in C(X)/OA . For, consider g ∈ fα ∩ β=α∈S fβ . Then n n g = fα · hα = i=1 rβi · fβi , where hα and each rβi are in C(X)/OA . Hence g · g = i=1 hα rβi fα fβi = 0. Thus g = 0. Therefore |S| ≤ a and hence S(A) ≤ a. 2 Corollary 4.8. The following statements hold. (1) dim(C(X)/M βX\X ) = S(βX \ X) ≤ dim(C(X)/CK (X)). (2) dim(C(X)/Socm C(X)) = S(βX \ I(X)) ≤ dim(C(X)/CF (X)). In [3, Remark 2.7], the authors have given an example to show that dim(C(X)/CF (X)) might be strictly greater than dimC(X). In the following, we obtain an easy proof for this result. Corollary 4.9. Let N be the set of positive integers with the discrete topology. Then dimC(N)dim(C(N)/ CF (N)). Proof. It is well known that CF (N) = M βN\N and S(βN \ N) = c = 2ℵ0 . By previous theorem, dim(C(N)/CF (N)) = c = 2ℵ0 . But dim(C(N)) = S(N) = ℵ0 . Hence the Goldie dimension of C(N) is strictly smaller than the Goldie dimension of C(N)/CF (N). 2 The following example shows that dim(C(X)/OA ) might be strictly greater than S(A). Example 4.10. Let X = N ∪ {ω} be the one point compactification of N. Then we have S(βX \ I(X)) = 1. By [9, 14F], Oω is not prime. Hence there are f, g ∈ C(X) such that f, g ∈ / Oω , but f g ∈ Oω . Thus f .g = 0
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and f , g are non-zero elements. This shows that the set {f , g } is an independent family of ideals of C(X)/Oω and hence dim(C(X)/Oω ) 1. Acknowledgement The authors would like to thank the referee for the careful reading of this paper and for pointing out some comments which led to a much improved paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F. Azarpanah, Essential ideals in C(X), Period. Math. Hung. 31 (1995) 105–112. F. Azarpanah, M. Ghirati, A. Taherifar, When is CF (X) = M βX\I(X) ?, Topol. Appl. 194 (2015) 22–25. F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 11 (2) (2008) 315–336. V. Camilo, W.K. Nicholson, M.F. Yousif, Ikeda–Nakayama rings, J. Algebra 226 (2000) 1001–1010. W.E. Dietrich, On the ideal structure of C(X), Trans. Am. Math. Soc. 152 (1970) 61–77. T. Dube, A note on the socle of certain types of f -rings, Bull. Iran. Math. Soc. 38 (2) (2012) 517–528. A.A. Estaji, O.A.S. Karamzadeh, On C(X) modulo its socle, Commun. Algebra 31 (2003) 1561–1571. M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. O.A.S. Karamzadeh, M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Am. Math. Soc. 93 (1) (1985) 179–184. M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. K. Samei, On the maximal spectrum of commutative semiprimitive rings, Colloq. Math. 83 (1) (2000) 5–13. A. Taherifar, Intersections of essential minimal prime ideals, Comment. Math. Univ. Carol. 55 (1) (2014) 121–130. A. Taherifar, Some new classes of topological spaces and annihilator ideals, Topol. Appl. 165 (2014) 84–97.