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Weakly clean rings
Journal of Algebra 401 (2014) 1–12
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Journal of Algebra www.elsevier.com/locate/jalgebra
Weakly clean rings Janez Šter Department of Mathematics, University of Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 15 March 2013 Available online 31 December 2013 Communicated by Louis Rowen MSC: 16U99 16S70 46L05 Keywords: Weakly clean ring Exchange ring Clean ring Corner ring π-regular ring C ∗ -algebra of real rank zero
a b s t r a c t We study the class of weakly clean rings which were introduced in [15]. It is known that weakly clean rings are a subclass of exchange rings and that they contain clean rings as a proper subclass. In this paper we prove that weakly clean rings also contain some other important examples of exchange rings, such as π-regular rings and C ∗ -algebras of real rank zero. Further, we prove that many classes of weakly clean rings can be viewed as corners of clean rings. This, for example, implies that every π-regular ring and every C ∗ -algebra of real rank zero is a corner of a clean ring. Lastly, we study the question when the ideal extension of weakly clean rings is weakly clean, and we give an example of a non-weakly clean exchange ring, answering the question in [15]. © 2013 Elsevier Inc. All rights reserved.
1. Introduction An element of a ring is called clean if it is a sum of an idempotent and a unit, and a ring is clean if its every element is clean. This notion was introduced by Nicholson in [12] where it was proved that every clean ring is an exchange ring and if idempotents in the ring are central then the converse also holds (see [12, Proposition 1.8]). Other examples of clean rings include semiperfect rings [7], unit-regular rings [5] and endomorphism rings of continuous modules [6]. The class of weakly clean rings was introduced in [15]. A ring R is said to be weakly clean if for every a ∈ R there exist an idempotent e ∈ R and a unit u ∈ R such that a − e − u ∈ (1 − e)Ra, or equivalently, the matrix ( a0 00 ) is a clean element of the ring 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.10.034
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M2 (R) (see [15, Proposition 2.2]). As shown in [15], every clean ring is weakly clean and every weakly clean ring is exchange. The well-known example due to Bergman is an example of a non-clean weakly clean ring (see [15, Example 3.1]). The question whether there exists a non-weakly clean exchange ring was left open in [15]. In [15], the notion of weakly clean rings was introduced with the purpose of answering the question whether corners of clean rings are clean. In this paper we study weakly clean rings as a separate subject of interest. The motivation for our interest comes from the obtained results. For example, we show that the class of weakly clean rings includes some important examples of exchange rings, such as π-regular rings and C ∗ -algebras of real rank zero. Further, we show that the class of weakly clean rings is closely related to the class of corners of clean rings. For example, we prove that for algebras over a field, and for rings with a nonzero characteristic, these two classes actually coincide. This, for example, implies that a unital C ∗ -algebra has real rank zero if and only if it is a corner of a clean ring. Also, every π-regular ring is a corner of a clean ring. In the last section we generalize the notion of the weakly clean property to non-unital rings. Inspired by Ara’s extension theorem for exchange rings [1, Theorem 2.2], we prove that, under some additional hypotheses, an analogous theorem holds also for weakly clean rings. In particular, we show that if R is a ring with an ideal I such that R/I is regular, I is weakly clean and idempotents lift modulo I, then R is weakly clean (Theorem 4.5). Also, we prove that if I and R/I are both π-regular rings then R is weakly clean (Proposition 4.9). We provide some examples showing that these statements cannot be generalized much further. These examples, in particular, are examples of non-weakly clean exchange rings, and thus they give a negative answer to the question in [15], asking whether or not every exchange ring is weakly clean. All rings in this paper will be non-commutative and unital, unless otherwise specified. Rings that do not necessarily have a unit will be called non-unital. For a ring R, we denote by Id(R), U (R), J(R) and Mn (R) the set of idempotents, the set of units, the Jacobson radical and the ring of n × n matrices over R, respectively. 2. Properties and examples First let us recall the definition of a weakly clean ring. Definition 2.1. (See [15, Definition 2.3].) Let R be a ring. An element a ∈ R is weakly clean in R if the following equivalent conditions hold: (i) There exist e ∈ Id(R) and u ∈ U (R) such that a − e − u ∈ (1 − e)Ra. (ii) There exist an idempotent e ∈ Ra and a unit u ∈ U (R) such that 1 − e = (1 − e)u(1 − a). (iii) The matrix ( a0 00 ) is clean in M2 (R). A ring R is weakly clean if every a ∈ R is weakly clean in R.
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Remark 2.2. The above conditions are equivalent by [15, Proposition 2.2 and Lemma 4.1]. In the following proposition we give some basic properties of weakly clean rings. Recall that a ring R is an exchange ring if for every a ∈ R there exists an idempotent e ∈ Ra such that 1 − e ∈ R(1 − a) (see [9,12]). Proposition 2.3. (i) Every clean ring is weakly clean and every weakly clean ring is exchange. (ii) Weakly clean rings are closed under homomorphic images, direct limits and (arbitrary) direct products. (iii) A ring R is weakly clean if and only if R/J(R) is weakly clean and idempotents lift modulo J(R). Proof. (i) follows from Definition 2.1 (ii) (also noted in [15, Remark 2.4]), (ii) can be verified directly, and (iii) is similar to the proof of [7, Proposition 7]. 2 Remark 2.4. The weakly clean property is left–right symmetric, meaning that a ∈ R is weakly clean in R if and only if it is weakly clean in the opposite ring Rop . This follows from Definition 2.1 (iii). However, we will see later that the weakly clean property is not symmetric in the sense that if a is weakly clean then 1 − a is weakly clean. This is in contrast to the clean and exchange properties which are symmetric in this sense. The following was proved in [15, Proposition 3.3]: Proposition 2.5. Let R be a ring and e ∈ Id(R). Then R is weakly clean if and only if the corner rings eRe and (1 − e)R(1 − e) are weakly clean. This proposition, in particular, implies that the weakly clean property is Morita invariant, and that the matrix ring Mn (R) is weakly clean if and only if the ring R is weakly clean, for every n. Note that the proof of [15, Proposition 3.3] also shows that if an element a ∈ eRe is weakly clean in R then it is also weakly clean in eRe. To give more examples of weakly clean rings, we will need the following useful lemma. Lemma 2.6. Let R be a ring and a ∈ R. If there exists e = e2 = 1 − f ∈ Ra such that f af is weakly clean in f Rf , then a is weakly clean in R. Proof. Let a ∈ R, e = e2 = 1−f ∈ Ra, and suppose that f af is weakly clean in f Rf , i.e. f af = g + u + (f − g)xf af with g ∈ Id(f Rf ), u ∈ U (f Rf ) and x ∈ f Rf . Note that e + u is invertible in R, and 1 + f ae is invertible, hence
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α = (1 + f ae)(e + u) = e + u + f ae is invertible in R. The equation gaf = g(f af ) = g + gu + g(f − g)xf af = g + gu gives gα = gu + gae = gu + ga − gaf = gu + ga − g − gu = ga − g. Therefore g(a − g − α) = 0, which gives a − g − α ∈ (1 − g)R. Furthermore, since af = a − ae ∈ Ra, we have g + u = f af − (f − g)xf af ∈ Ra, therefore g + α = g + e + u + f ae ∈ Ra. It follows that a − g − α ∈ Ra and hence a − g − α ∈ (1 − g)Ra, as desired. 2 Now we are ready to give some more examples of weakly clean rings. Example 2.7. (1) An element of a ring a ∈ R is called π-regular if there exist r ∈ R and n 1 such that an = an ran , and R is π-regular if every a ∈ R is π-regular. It is known that π-regular rings are exchange (see [16, Example 2.3]). Let us prove that in fact they are weakly clean. Let R be π-regular and a ∈ R. By assumption, there exist n 1 and r ∈ R such that an ran = an . Writing e = ran ∈ Ra and f = 1 − e, we can easily check that (af )n = 0. Therefore f af ∈ f Rf is a nilpotent element and hence clean in f Rf . By Lemma 2.6 it follows that a is weakly clean in R, as desired. (2) A ring R is (von Neumann) regular if for every a ∈ R there exists r ∈ R such that a = ara, and R is semiregular if R/J(R) is regular and idempotents can be lifted modulo J(R). Every semiregular ring is exchange [12, Proposition 1.6]. In fact, by (1) and Proposition 2.3 (iii) every semiregular ring is weakly clean. (3) A (complex unital) C ∗ -algebra A has real rank zero [3] if for every x = x∗ ∈ A and ε > 0 there exists an invertible element u = u∗ ∈ A such that x − u < ε. In [2, Theorem 7.2] Ara et al. proved that a unital C ∗ -algebra is an exchange ring if and only if it has real rank zero. Let us show that in fact every exchange unital C ∗ -algebra A is weakly clean. Take a ∈ A. Following the proof of [2, Theorem 7.2 (a)⇒(b)], we see that A satisfies the hypothesis of [2, Lemma 7.1], hence there exists an idempotent e = 1 − f ∈ Aa such that af < 1. Thus 1 − af is invertible, which clearly implies that f − f af is invertible in f Af . Thus f af is clean in f Af , and hence by Lemma 2.6 a is weakly clean in A, as desired. 3. Corners of clean rings In [15, Example 3.4] it was shown that every weakly clean ring R with char(R) = 2 is a corner of a clean ring, i.e., there exist a clean ring S and an idempotent e ∈ S such that R ∼ = eSe. We will generalize this result to the case when R is any algebra over a field, or any ring with char(R) = 0.
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To prove the main result, we will need a few lemmas. Lemma 3.1. Let R be a ring and a, λ ∈ R elements such that λ ∈ U (R) and 1−λ ∈ U (R). Then the matrix ( a0 λ0 ) is clean in M2 (R). 1−λ λ −λ Proof. Take E = ( 1−λ ) ∈ M2 (R) and U = ( a−1+λ ) ∈ M2 (R). Then E is clearly an λ −1+λ 0 a 0 idempotent and U a unit in M2 (R), with ( 0 λ ) = E + U . 2
Lemma 3.2. Let R be an exchange ring and A = ( ac db ) ∈ M2 (R) a matrix such that a is weakly clean in R and b, c, d ∈ J(R). Then A is clean in M2 (R). Proof. For every x ∈ R, let x denote the homomorphic image of x in R = R/J(R). First observe that a is weakly clean in R, hence A = ( a0 00 ) is clean in M2 (R). Write A = + μ, with ∈ Id(M2 (R)) and μ ∈ U (M2 (R)). Since R is an exchange ring, there exists E ∈ Id(M2 (R)) such that E = . Thus we have A − E = μ ∈ U (M2 (R)) and hence A − E ∈ U (M2 (R)), as desired. 2 Recall that a ring R is strongly π-regular if for every a ∈ R there exist n 1 and r ∈ R such that an = an+1 r. Every commutative π-regular ring is strongly π-regular. Every strongly π-regular ring is π-regular (see [13]). Also, every strongly π-regular ring is clean [13, Theorem 1]. The following characterization of strongly π-regular rings is well-known: Lemma 3.3. (See [13, equivalence (1)⇔(4) on p. 3589].) A ring R is strongly π-regular if and only if for every a ∈ R there exists e ∈ Id(R) such that ea = ae, ae is a nilpotent in eRe and a(1 − e) is a unit in (1 − e)R(1 − e). With this lemma we can easily prove: Lemma 3.4. A ring R is strongly π-regular if and only if for every a ∈ R there exist pair-wise orthogonal idempotents e1 , e2 , e3 ∈ Id(R) with e1 + e2 + e3 = 1, such that ei a = aei for each i, ae1 is a nilpotent in e1 Re1 , e2 − ae2 is a nilpotent in e2 Re2 , and ae3 , e3 − ae3 ∈ U (e3 Re3 ). Proof. Direction (⇐) is clear from Lemma 3.3. Thus, let R be strongly π-regular and a ∈ R. By Lemma 3.3 we have e1 = 1 − f1 ∈ Id(R) such that e1 a = ae1 , ae1 is a nilpotent in e1 Re1 and af1 is a unit in f1 Rf1 . Note that corners of strongly π-regular rings are strongly π-regular. Hence, applying Lemma 3.3 for the ring f1 Rf1 and the element α = f1 − af1 ∈ f1 Rf1 , we have e2 = f1 − e3 ∈ Id(f1 Rf1 ) such that e2 α = αe2 , αe2 = e2 − ae2 is a nilpotent in e2 Re2 , and αe3 = e3 − ae3 is a unit in e3 Re3 . Note that, since af1 ∈ U (f1 Rf1 ) commutes with e3 ∈ Id(f1 Rf1 ), ae3 is a unit in e3 Re3 . Furthermore, the equation e2 α = αe2 implies that e2 a = ae2 and e3 a = ae3 . Thus idempotents e1 , e2 , e3 indeed fulfill all the desired properties. 2
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Now we are ready to give the main result. For a ring R, we denote by Z(R) the center of R. Theorem 3.5. Let R be a ring such that Z(R) contains a π-regular subring k. Then R is weakly clean if and only if it is a corner of a clean algebra over k. Proof. Direction (⇐) is clear since the class of weakly clean rings is closed under taking corners. Thus, assume that R is weakly clean. To obtain a clean ring S with R as a corner, we will perform a construction similar to that in [15, Example 3.4]. Let MN (R) denote the ring of countably infinite matrices over R with finite columns. Let T ⊆ MN (R) be a non-unital subring consisting of all matrices with only finitely many nonzero entries. Define S = T + k · 1. Then S is a unital algebra over k. As in [15, Example 3.4], we see that R is a corner ring of S. Thus, it remains to show that S is clean. Take an element ⎛ ⎞ A0 ⎜ ⎟ λ ⎜ ⎟ ⎟ ∈ S, A=⎜ ⎜ ⎟ λ ⎝ ⎠ .. . with A0 ∈ Mn (R) and λ ∈ k. Note that k is a commutative strongly π-regular ring. Hence by Lemma 3.4 there exist pair-wise orthogonal idempotents e1 , e2 , e3 ∈ k, with ei = 1, such that λe1 is a nilpotent in ke1 , e2 − λe2 is a nilpotent in ke2 and λe3 , e3 − λe3 are units in ke3 . Note that e1 , e2 , e3 are central idempotents in R, hence R ∼ = Re1 × Re2 × Re3 . This ∼ isomorphism induces an isomorphism S = Se1 ×Se2 ×Se3 . According to this isomorphism the matrix A ∈ S corresponds to the triple (Ae1 , Ae2 , Ae3 ). To prove that A is clean in S, we need to prove that each Aei is clean in Sei . First, note that λe1 is a central nilpotent in Re1 , hence λe1 ∈ J(Re1 ). The ring Re1 is weakly clean since it is a homomorphic image of R. Thus, by Lemma 3.2, the matrix A0 e1 0
is clean in M2n (Re1 ) (with In denoting the n × n identity matrix). Hence 0 In λe1 Ae1 is clean in Se1 . Similarly, e2 − λe2 ∈ Re2 is a central nilpotent and hence e2 − λe2 ∈ J(Re2 ). As Re2 I e −A e
0 is weakly clean, by Lemma 3.2 it follows that the matrix n 2 0 0 2 In (e2 −λe2 ) is clean in M2n (Re2 ), hence 0 0 In e2 − A0 e2 A0 e2 I2n e2 − = 0 In (e2 − λe2 ) 0 In λe2 is clean in M2n (Re2 ). Hence Ae2 is clean in Se2 .
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Lastly, since λe3 and e3 − λe3 are invertible in Re3 , by Lemma 3.1 the matrix A0 e3 0
is clean in M2n (Re3 ). Thus Ae3 is clean in Se3 . We conclude that each 0 In λe3 matrix Aei is clean in Sei , and hence A is clean in S. This completes the proof of the theorem. 2 Now we draw some consequences of the theorem. Corollary 3.6. Let R be an algebra over a field F . Then R is weakly clean if and only if it is a corner ring of a clean F -algebra. Proof. Apply Theorem 3.5 with k = F .
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We do not know if every unital C ∗ -algebra with real rank zero is clean. However, by Corollary 3.6 we have: Corollary 3.7. A complex unital C ∗ -algebra has real rank zero if and only if it is a corner ring of a clean C-algebra. 2 Corollary 3.8. Let R be a ring with char(R) = 0. Then R is weakly clean if and only if it is a corner ring of a clean ring. Proof. If char(R) = n = 0 then Z(R) contains a π-regular subring Zn = Z/nZ. Hence we may apply Theorem 3.5 with k = Zn . 2 The center of a π-regular ring is π-regular (see [10, Theorem 1]). Thus, by Theorem 3.5, we have: Corollary 3.9. Every π-regular ring is a corner of a clean ring.
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In light of the above results, it is natural to ask: Question 3.10. Is every weakly clean ring a corner ring of some clean ring? The affirmative answer to this question would give a characterization of weakly clean rings as the class of corners of clean rings. We conclude by noting that the center of a clean ring need not be clean (see [4, Proposition 2.5]). Thus the methods of Theorem 3.5 seemingly cannot be applied for arbitrary weakly clean rings. 4. Extensions of weakly clean rings Ara in [1] defined the notion of a non-unital exchange ring and proved the extension theorem for exchange rings. We will prove a similar result (with some more assumptions)
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for weakly clean rings. First, let us introduce the notion of a weakly clean non-unital ring. Definition 4.1. Let I be a non-unital ring and a ∈ I. Then a is called weakly clean in I if it can be written as a = e + p + exa for some e ∈ Id(I), x ∈ I, and p an element of the set Q(I) = {p ∈ I | there exists q ∈ I such that p + q + pq = p + q + qp = 0}. A ring I is weakly clean if each a ∈ I is weakly clean in I. Remark 4.2. (i) A simple verification shows that the above definition matches with the standard definition if I has a unity. Next, every non-unital clean ring (in the sense of Nicholson and Zhou [14]) is weakly clean, and every weakly clean non-unital ring is exchange (in the sense of Ara [1]). (ii) If I is a non-unital subring of a ring R such that I is weakly clean, then every a ∈ I is weakly clean (in the unital sense) in R. Indeed, if a ∈ I, then −a ∈ I and by assumption, −a = e + p + exa with e = e2 ∈ I, p ∈ Q(I) and x ∈ I. It follows that a = (1 − e) − (1 + p) − exa, with 1 − e ∈ Id(R) and 1 + p ∈ U (R). (iii) Conversely, if I is an ideal of a unital ring R such that every a ∈ I is weakly clean in R, then I is a weakly clean ring. Indeed, for any a ∈ I, −a is weakly clean in R, i.e. −a = e + u + f xa, with e = 1 − f ∈ Id(R), u ∈ U (R) and x ∈ R. We have e + u ∈ I, hence f u ∈ I and f ∈ I. Thus a = f − (1 + u) − f · f x · a, with −1 − u ∈ Q(I) and f x ∈ I. (iv) The above conclusion fails if I is only a subring (not an ideal). For example, take I = Z and R = Q; then every a ∈ I is clean in R, but I is not weakly clean. (v) The non-unital weakly clean property is left–right symmetric. The easiest way to prove this is through observation that an element a of a non-unital ring I is weakly clean in I if and only if (0, −a) is a weakly clean element of the unitization I 1 = Z ⊕ I of I. The left–right symmetry then follows from the unital left–right symmetry. Example 4.3. (1) A (possibly non-unital) C ∗ -algebra is weakly clean if and only if it has real rank zero. Indeed, if A is a weakly clean non-unital C ∗ -algebra, then by Remark 4.2 (i) A is exchange, hence by [1, Theorem 3.8] it has real rank zero. Conversely, if A has real rank zero, then, by definition, its unitization A1 = C ⊕ A has real rank zero (see [3]) and is therefore weakly clean by Example 2.7 (3). A is an ideal of A1 , hence by Remark 4.2 (iii), A is also weakly clean. (2) Every non-unital π-regular ring is weakly clean (this can be easily verified by adapting the proof of the unital case). Recall that if R is a (unital) ring, e ∈ R an idempotent and I an ideal of R, then eIe = {exe | x ∈ I} is an ideal of the corner ring eRe.
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Lemma 4.4. Let I be an ideal of a ring R and e ∈ Id(R). If I is weakly clean then eIe is weakly clean. Proof. Suppose that I is weakly clean and take a ∈ eIe. Since a is weakly clean in I, by Remark 4.2 (ii) it is also weakly clean in R and hence weakly clean in eRe. Thus every a ∈ eIe is weakly clean in eRe. By Remark 4.2 (iii) it follows that eIe is weakly clean. 2 Recall that by Example 2.7 (1), regular rings are weakly clean. The following theorem extends this result: Theorem 4.5. Let R be a ring and I an ideal such that I is weakly clean, R/I is regular and idempotents lift modulo I. Then R is weakly clean. Proof. In the proof we will use techniques similar to those in [1, Proof of Theorem 2.2]. Let R and I satisfy the assumptions of the theorem. For any x ∈ R, we denote by x the homomorphic image of x in R = R/I. Take any element a ∈ R, and denote α = a. Since R is regular, there exists β ∈ R such that αβα = α. Since αβ is an idempotent in R, and idempotents lift modulo I, there exists e ∈ Id(R) such that e = αβ. Write β = b for some b ∈ R. Observe that eabe − e = 0, and therefore eabe − e ∈ I ∩ eRe = eIe. Now, since eIe is weakly clean, there exist g ∈ Id(eIe), p ∈ Q(eIe) and x ∈ eIe such that eabe − e = g + p + gx(eabe − e). Writing u = e + p ∈ U (eRe), we have eabe = g + u + gx(eabe − e).
(1)
Let v denote the inverse of u in eRe, and h = e − g. Multiplying (1) by vh from the left, we have vhabe = vhu. It follows that (bvha)2 = bvhuvha = bvha, and hence q = bvha is an idempotent in Ra. By Lemma 2.6, it suffices to see that (1 − q)a(1 − q) is weakly clean in (1 − q)R(1 − q). Observe that u = e, hence v = e and q = βv(e − g)α = βeα = βα. Thus we have (1 − q)a(1 − q) = (1−βα)α(1−βα) = 0. We conclude that (1−q)a(1−q) ∈ (1 − q)R(1 − q) ∩ I = (1 − q)I(1 − q). Since (1 − q)I(1 − q) is weakly clean, the theorem is proved. 2 Theorem 4.5 cannot be generalized to the case when R/I is semiregular or even local: Example 4.6. There exists a ring R with an ideal I such that I is a clean non-unital ring, idempotents lift modulo I and R /I is local, but R is not weakly clean. Let I and R be as in [15, Example 3.1]. Following notation in [15], we denote by ψ : R → F ((X)) a surjective ring homomorphism with the kernel I, where F ((X)) denotes the field of formal Laurent series over the field F . Note that R is a weakly clean ring. Now, let k ⊆ F ((X)) be a subring of all rational functions p(X)q(X)−1 , with p(X), q(X) ∈ F [X] and q(X) = 0, such that q(X) is not divisible by X − 1. Observe
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that k is a local ring since it is isomorphic to the localization of the polynomial ring F [X] at the prime ideal F [X](X − 1). The Jacobson radical of this ring is precisely J(k) = k(X − 1). Now define
R = A ∈ R | ψ(A) ∈ k . The quotient mapping ψ : R → F ((X)) induces a surjection ψ|R : R → k with the kernel I. Clearly, idempotents in k lift modulo I. The ring I is clean (cf. [15, Proposition 4.2]). It remains to show that R is not weakly clean. Let A ∈ R denote the left shift operator, so that ψ(A) = X. Let us prove that 1 − A is not weakly clean in R . Suppose, to the contrary, that E = T (1 − A) and 1−E = (1−E)U A, with E ∈ Id(R ), T ∈ R and U ∈ U (R ). Since ψ(E) is an idempotent and ψ(E) = ψ(T )(1 − X) ∈ J(k), it follows that ψ(E) = 0. Hence ψ(1 − E) = 1, which gives 1 = ψ(U )ψ(A) and ψ(U ) = ψ(A)−1 = X −1 . But this contradicts the fact that U is invertible in R. Hence 1 − A is not weakly clean. Remark 4.7. (i) The ring in the above example is an exchange ring by [1] (I and R /I are exchange rings and idempotents lift modulo I). Thus we answer in negative the question in [15], asking whether every exchange ring is weakly clean. (ii) Note that the element A in the above example is weakly clean but 1 − A is not. Thus, this example also shows that the weakly clean property is not symmetric in the sense that if a is weakly clean then 1 − a is weakly clean. In Example 4.6, the fact that R is not weakly clean follows from the nonzero radical of the factor ring R /I. The following example shows that the nonzero radical of the factor ring is not crucial. Example 4.8. There exists a ring S with an ideal J such that J is clean, S/J is clean with J(S/J) = 0 and idempotents lift modulo J, but S is not weakly clean. Let I, R and R be as in Example 4.6. We repeat the construction performed in [12, Example 1.7]. Define
S = (a1 , . . . , an , a, a, . . .) | ai ∈ R and a ∈ R and
J = (a1 , . . . , an , a, a, . . .) | ai , a ∈ I . Then S is a ring with an ideal J such that
S/J ∼ = (t1 , . . . , tn , t, t, . . .) | ti ∈ F (X) and t ∈ k .
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Observe that J and S/J are clean rings, J(S/J) = 0 and idempotents lift modulo J, but S is not weakly clean since R is its homomorphic image. We do not know whether Theorem 4.5 can be extended to the case when R/I is π-regular or even artinian. Note that the factor rings R /I and S/J in the above examples are not π-regular. However, we are able to give the following partial answer to this question. Proposition 4.9. Let R be a ring with an ideal I such that I and R/I are both π-regular rings. Then R is weakly clean. Note that in this proposition we have no assumption that idempotents lift modulo I. Idempotent lifting indeed comes automatically: Lemma 4.10. (See [11, Lemma 3].) If R is any ring and I an ideal of R such that I is π-regular then idempotents lift modulo I. Proof of Proposition 4.9. Suppose that R is a ring and I an ideal such that I and R/I are π-regular rings. For any x ∈ R, we denote by x the homomorphic image of x in R = R/I. Take any element a ∈ R, and denote α = a. Since R is π-regular, there exist β ∈ R and n 1 such that αn βαn = αn . Since αn β is an idempotent in R, by Lemma 4.10 there exists e ∈ Id(R) such that e = αn β. Write β = b for some b ∈ R. Observe that ean be − e = 0 and therefore ean be − e ∈ I ∩ eRe = eIe. Since eIe is weakly clean, there exist g ∈ Id(eIe), p ∈ Q(eIe) and x ∈ eIe such that ean be − e = g + p + gx(ean be − e). Writing u = e + p ∈ U (eRe) we have
ean be = g + u + gx ean be − e . Let v denote the inverse of u in eRe, and h = e − g. Multiplying the above equation by vh from the left, we have vhan be = vhu. It follows that (bvhan )2 = bvhuvhan = bvhan , and hence q = bvhan is an idempotent in Ra. By Lemma 2.6, it suffices to prove that (1 − q)a(1 − q) is weakly clean in (1 − q)R(1 − q). Furthermore, by the note following Proposition 2.5 it suffices to prove that (1 − q)a(1 − q) is weakly clean in R. Observe that u = e, hence v = e and q = βv(e − g)αn = βeαn = βαn . As in Example 2.7 (1) we see that (α(1 − βαn ))n = 0. Hence (a(1 − q))n ∈ I and therefore ((1−q)a(1−q))n ∈ I. Since I is π-regular, it follows that ((1−q)a(1−q))n is π-regular in I and hence (1−q)a(1−q) ∈ R is π-regular in R. Hence by Example 2.7 (1) (1−q)a(1−q) is weakly clean in R. This completes the proof. 2 It is a natural question if the assumptions of Proposition 4.9 actually imply that the ring R is π-regular. In this direction we know only one partial result by Fuchs and Rangaswamy:
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Lemma 4.11. (See [8, Lemma 1].) Let R be a ring with an ideal I such that I is π-regular and for every a ∈ R there exist x ∈ R and n 1 with ax = xa and an xan − an ∈ I. Then R is π-regular. This result suggests that the answer in general is negative. Thus, Proposition 4.9 probably presents a non-trivial generalization of Example 2.7 (1). Acknowledgment I would like to thank Slovenian Research Agency for financial support. References [1] P. Ara, Extensions of exchange rings, J. Algebra 197 (2) (1997) 409–423. [2] P. Ara, K.R. Goodearl, K.C. O’Meara, E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998) 105–137. [3] L.G. Brown, G.K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1) (1991) 131–149. [4] W.D. Burgess, R. Raphael, On embedding rings in clean rings, Comm. Algebra 41 (2) (2013) 552–564. [5] V.P. Camillo, D. Khurana, A characterization of unit regular rings, Comm. Algebra 29 (5) (2001) 2293–2295. [6] V.P. Camillo, D. Khurana, T.Y. Lam, W.K. Nicholson, Y. Zhou, Continuous modules are clean, J. Algebra 304 (1) (2006) 94–111. [7] V.P. Camillo, H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (12) (1994) 4737–4749. [8] L. Fuchs, K.M. Rangaswamy, On generalized regular rings, Math. Z. 107 (1968) 71–81. [9] K.R. Goodearl, R.B. Warfield Jr., Algebras over zero-dimensional rings, Math. Ann. 223 (2) (1976) 157–168. [10] N.H. McCoy, Generalized regular rings, Bull. Amer. Math. Soc. 45 (2) (1939) 175–178. [11] P. Menal, On π-regular rings whose primitive factor rings are Artinian, J. Pure Appl. Algebra 20 (1) (1981) 71–78. [12] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278. [13] W.K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra 27 (8) (1999) 3583–3592. [14] W.K. Nicholson, Y. Zhou, Clean general rings, J. Algebra 291 (1) (2005) 297–311. [15] J. Šter, Corner rings of a clean ring need not be clean, Comm. Algebra 40 (5) (2012) 1595–1604. [16] J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (2) (1986) 437–453.
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Journal of Algebra www.elsevier.com/locate/jalgebra
Weakly clean rings Janez Šter Department of Mathematics, University of Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 15 March 2013 Available online 31 December 2013 Communicated by Louis Rowen MSC: 16U99 16S70 46L05 Keywords: Weakly clean ring Exchange ring Clean ring Corner ring π-regular ring C ∗ -algebra of real rank zero
a b s t r a c t We study the class of weakly clean rings which were introduced in [15]. It is known that weakly clean rings are a subclass of exchange rings and that they contain clean rings as a proper subclass. In this paper we prove that weakly clean rings also contain some other important examples of exchange rings, such as π-regular rings and C ∗ -algebras of real rank zero. Further, we prove that many classes of weakly clean rings can be viewed as corners of clean rings. This, for example, implies that every π-regular ring and every C ∗ -algebra of real rank zero is a corner of a clean ring. Lastly, we study the question when the ideal extension of weakly clean rings is weakly clean, and we give an example of a non-weakly clean exchange ring, answering the question in [15]. © 2013 Elsevier Inc. All rights reserved.
1. Introduction An element of a ring is called clean if it is a sum of an idempotent and a unit, and a ring is clean if its every element is clean. This notion was introduced by Nicholson in [12] where it was proved that every clean ring is an exchange ring and if idempotents in the ring are central then the converse also holds (see [12, Proposition 1.8]). Other examples of clean rings include semiperfect rings [7], unit-regular rings [5] and endomorphism rings of continuous modules [6]. The class of weakly clean rings was introduced in [15]. A ring R is said to be weakly clean if for every a ∈ R there exist an idempotent e ∈ R and a unit u ∈ R such that a − e − u ∈ (1 − e)Ra, or equivalently, the matrix ( a0 00 ) is a clean element of the ring 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.10.034
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M2 (R) (see [15, Proposition 2.2]). As shown in [15], every clean ring is weakly clean and every weakly clean ring is exchange. The well-known example due to Bergman is an example of a non-clean weakly clean ring (see [15, Example 3.1]). The question whether there exists a non-weakly clean exchange ring was left open in [15]. In [15], the notion of weakly clean rings was introduced with the purpose of answering the question whether corners of clean rings are clean. In this paper we study weakly clean rings as a separate subject of interest. The motivation for our interest comes from the obtained results. For example, we show that the class of weakly clean rings includes some important examples of exchange rings, such as π-regular rings and C ∗ -algebras of real rank zero. Further, we show that the class of weakly clean rings is closely related to the class of corners of clean rings. For example, we prove that for algebras over a field, and for rings with a nonzero characteristic, these two classes actually coincide. This, for example, implies that a unital C ∗ -algebra has real rank zero if and only if it is a corner of a clean ring. Also, every π-regular ring is a corner of a clean ring. In the last section we generalize the notion of the weakly clean property to non-unital rings. Inspired by Ara’s extension theorem for exchange rings [1, Theorem 2.2], we prove that, under some additional hypotheses, an analogous theorem holds also for weakly clean rings. In particular, we show that if R is a ring with an ideal I such that R/I is regular, I is weakly clean and idempotents lift modulo I, then R is weakly clean (Theorem 4.5). Also, we prove that if I and R/I are both π-regular rings then R is weakly clean (Proposition 4.9). We provide some examples showing that these statements cannot be generalized much further. These examples, in particular, are examples of non-weakly clean exchange rings, and thus they give a negative answer to the question in [15], asking whether or not every exchange ring is weakly clean. All rings in this paper will be non-commutative and unital, unless otherwise specified. Rings that do not necessarily have a unit will be called non-unital. For a ring R, we denote by Id(R), U (R), J(R) and Mn (R) the set of idempotents, the set of units, the Jacobson radical and the ring of n × n matrices over R, respectively. 2. Properties and examples First let us recall the definition of a weakly clean ring. Definition 2.1. (See [15, Definition 2.3].) Let R be a ring. An element a ∈ R is weakly clean in R if the following equivalent conditions hold: (i) There exist e ∈ Id(R) and u ∈ U (R) such that a − e − u ∈ (1 − e)Ra. (ii) There exist an idempotent e ∈ Ra and a unit u ∈ U (R) such that 1 − e = (1 − e)u(1 − a). (iii) The matrix ( a0 00 ) is clean in M2 (R). A ring R is weakly clean if every a ∈ R is weakly clean in R.
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Remark 2.2. The above conditions are equivalent by [15, Proposition 2.2 and Lemma 4.1]. In the following proposition we give some basic properties of weakly clean rings. Recall that a ring R is an exchange ring if for every a ∈ R there exists an idempotent e ∈ Ra such that 1 − e ∈ R(1 − a) (see [9,12]). Proposition 2.3. (i) Every clean ring is weakly clean and every weakly clean ring is exchange. (ii) Weakly clean rings are closed under homomorphic images, direct limits and (arbitrary) direct products. (iii) A ring R is weakly clean if and only if R/J(R) is weakly clean and idempotents lift modulo J(R). Proof. (i) follows from Definition 2.1 (ii) (also noted in [15, Remark 2.4]), (ii) can be verified directly, and (iii) is similar to the proof of [7, Proposition 7]. 2 Remark 2.4. The weakly clean property is left–right symmetric, meaning that a ∈ R is weakly clean in R if and only if it is weakly clean in the opposite ring Rop . This follows from Definition 2.1 (iii). However, we will see later that the weakly clean property is not symmetric in the sense that if a is weakly clean then 1 − a is weakly clean. This is in contrast to the clean and exchange properties which are symmetric in this sense. The following was proved in [15, Proposition 3.3]: Proposition 2.5. Let R be a ring and e ∈ Id(R). Then R is weakly clean if and only if the corner rings eRe and (1 − e)R(1 − e) are weakly clean. This proposition, in particular, implies that the weakly clean property is Morita invariant, and that the matrix ring Mn (R) is weakly clean if and only if the ring R is weakly clean, for every n. Note that the proof of [15, Proposition 3.3] also shows that if an element a ∈ eRe is weakly clean in R then it is also weakly clean in eRe. To give more examples of weakly clean rings, we will need the following useful lemma. Lemma 2.6. Let R be a ring and a ∈ R. If there exists e = e2 = 1 − f ∈ Ra such that f af is weakly clean in f Rf , then a is weakly clean in R. Proof. Let a ∈ R, e = e2 = 1−f ∈ Ra, and suppose that f af is weakly clean in f Rf , i.e. f af = g + u + (f − g)xf af with g ∈ Id(f Rf ), u ∈ U (f Rf ) and x ∈ f Rf . Note that e + u is invertible in R, and 1 + f ae is invertible, hence
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α = (1 + f ae)(e + u) = e + u + f ae is invertible in R. The equation gaf = g(f af ) = g + gu + g(f − g)xf af = g + gu gives gα = gu + gae = gu + ga − gaf = gu + ga − g − gu = ga − g. Therefore g(a − g − α) = 0, which gives a − g − α ∈ (1 − g)R. Furthermore, since af = a − ae ∈ Ra, we have g + u = f af − (f − g)xf af ∈ Ra, therefore g + α = g + e + u + f ae ∈ Ra. It follows that a − g − α ∈ Ra and hence a − g − α ∈ (1 − g)Ra, as desired. 2 Now we are ready to give some more examples of weakly clean rings. Example 2.7. (1) An element of a ring a ∈ R is called π-regular if there exist r ∈ R and n 1 such that an = an ran , and R is π-regular if every a ∈ R is π-regular. It is known that π-regular rings are exchange (see [16, Example 2.3]). Let us prove that in fact they are weakly clean. Let R be π-regular and a ∈ R. By assumption, there exist n 1 and r ∈ R such that an ran = an . Writing e = ran ∈ Ra and f = 1 − e, we can easily check that (af )n = 0. Therefore f af ∈ f Rf is a nilpotent element and hence clean in f Rf . By Lemma 2.6 it follows that a is weakly clean in R, as desired. (2) A ring R is (von Neumann) regular if for every a ∈ R there exists r ∈ R such that a = ara, and R is semiregular if R/J(R) is regular and idempotents can be lifted modulo J(R). Every semiregular ring is exchange [12, Proposition 1.6]. In fact, by (1) and Proposition 2.3 (iii) every semiregular ring is weakly clean. (3) A (complex unital) C ∗ -algebra A has real rank zero [3] if for every x = x∗ ∈ A and ε > 0 there exists an invertible element u = u∗ ∈ A such that x − u < ε. In [2, Theorem 7.2] Ara et al. proved that a unital C ∗ -algebra is an exchange ring if and only if it has real rank zero. Let us show that in fact every exchange unital C ∗ -algebra A is weakly clean. Take a ∈ A. Following the proof of [2, Theorem 7.2 (a)⇒(b)], we see that A satisfies the hypothesis of [2, Lemma 7.1], hence there exists an idempotent e = 1 − f ∈ Aa such that af < 1. Thus 1 − af is invertible, which clearly implies that f − f af is invertible in f Af . Thus f af is clean in f Af , and hence by Lemma 2.6 a is weakly clean in A, as desired. 3. Corners of clean rings In [15, Example 3.4] it was shown that every weakly clean ring R with char(R) = 2 is a corner of a clean ring, i.e., there exist a clean ring S and an idempotent e ∈ S such that R ∼ = eSe. We will generalize this result to the case when R is any algebra over a field, or any ring with char(R) = 0.
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To prove the main result, we will need a few lemmas. Lemma 3.1. Let R be a ring and a, λ ∈ R elements such that λ ∈ U (R) and 1−λ ∈ U (R). Then the matrix ( a0 λ0 ) is clean in M2 (R). 1−λ λ −λ Proof. Take E = ( 1−λ ) ∈ M2 (R) and U = ( a−1+λ ) ∈ M2 (R). Then E is clearly an λ −1+λ 0 a 0 idempotent and U a unit in M2 (R), with ( 0 λ ) = E + U . 2
Lemma 3.2. Let R be an exchange ring and A = ( ac db ) ∈ M2 (R) a matrix such that a is weakly clean in R and b, c, d ∈ J(R). Then A is clean in M2 (R). Proof. For every x ∈ R, let x denote the homomorphic image of x in R = R/J(R). First observe that a is weakly clean in R, hence A = ( a0 00 ) is clean in M2 (R). Write A = + μ, with ∈ Id(M2 (R)) and μ ∈ U (M2 (R)). Since R is an exchange ring, there exists E ∈ Id(M2 (R)) such that E = . Thus we have A − E = μ ∈ U (M2 (R)) and hence A − E ∈ U (M2 (R)), as desired. 2 Recall that a ring R is strongly π-regular if for every a ∈ R there exist n 1 and r ∈ R such that an = an+1 r. Every commutative π-regular ring is strongly π-regular. Every strongly π-regular ring is π-regular (see [13]). Also, every strongly π-regular ring is clean [13, Theorem 1]. The following characterization of strongly π-regular rings is well-known: Lemma 3.3. (See [13, equivalence (1)⇔(4) on p. 3589].) A ring R is strongly π-regular if and only if for every a ∈ R there exists e ∈ Id(R) such that ea = ae, ae is a nilpotent in eRe and a(1 − e) is a unit in (1 − e)R(1 − e). With this lemma we can easily prove: Lemma 3.4. A ring R is strongly π-regular if and only if for every a ∈ R there exist pair-wise orthogonal idempotents e1 , e2 , e3 ∈ Id(R) with e1 + e2 + e3 = 1, such that ei a = aei for each i, ae1 is a nilpotent in e1 Re1 , e2 − ae2 is a nilpotent in e2 Re2 , and ae3 , e3 − ae3 ∈ U (e3 Re3 ). Proof. Direction (⇐) is clear from Lemma 3.3. Thus, let R be strongly π-regular and a ∈ R. By Lemma 3.3 we have e1 = 1 − f1 ∈ Id(R) such that e1 a = ae1 , ae1 is a nilpotent in e1 Re1 and af1 is a unit in f1 Rf1 . Note that corners of strongly π-regular rings are strongly π-regular. Hence, applying Lemma 3.3 for the ring f1 Rf1 and the element α = f1 − af1 ∈ f1 Rf1 , we have e2 = f1 − e3 ∈ Id(f1 Rf1 ) such that e2 α = αe2 , αe2 = e2 − ae2 is a nilpotent in e2 Re2 , and αe3 = e3 − ae3 is a unit in e3 Re3 . Note that, since af1 ∈ U (f1 Rf1 ) commutes with e3 ∈ Id(f1 Rf1 ), ae3 is a unit in e3 Re3 . Furthermore, the equation e2 α = αe2 implies that e2 a = ae2 and e3 a = ae3 . Thus idempotents e1 , e2 , e3 indeed fulfill all the desired properties. 2
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Now we are ready to give the main result. For a ring R, we denote by Z(R) the center of R. Theorem 3.5. Let R be a ring such that Z(R) contains a π-regular subring k. Then R is weakly clean if and only if it is a corner of a clean algebra over k. Proof. Direction (⇐) is clear since the class of weakly clean rings is closed under taking corners. Thus, assume that R is weakly clean. To obtain a clean ring S with R as a corner, we will perform a construction similar to that in [15, Example 3.4]. Let MN (R) denote the ring of countably infinite matrices over R with finite columns. Let T ⊆ MN (R) be a non-unital subring consisting of all matrices with only finitely many nonzero entries. Define S = T + k · 1. Then S is a unital algebra over k. As in [15, Example 3.4], we see that R is a corner ring of S. Thus, it remains to show that S is clean. Take an element ⎛ ⎞ A0 ⎜ ⎟ λ ⎜ ⎟ ⎟ ∈ S, A=⎜ ⎜ ⎟ λ ⎝ ⎠ .. . with A0 ∈ Mn (R) and λ ∈ k. Note that k is a commutative strongly π-regular ring. Hence by Lemma 3.4 there exist pair-wise orthogonal idempotents e1 , e2 , e3 ∈ k, with ei = 1, such that λe1 is a nilpotent in ke1 , e2 − λe2 is a nilpotent in ke2 and λe3 , e3 − λe3 are units in ke3 . Note that e1 , e2 , e3 are central idempotents in R, hence R ∼ = Re1 × Re2 × Re3 . This ∼ isomorphism induces an isomorphism S = Se1 ×Se2 ×Se3 . According to this isomorphism the matrix A ∈ S corresponds to the triple (Ae1 , Ae2 , Ae3 ). To prove that A is clean in S, we need to prove that each Aei is clean in Sei . First, note that λe1 is a central nilpotent in Re1 , hence λe1 ∈ J(Re1 ). The ring Re1 is weakly clean since it is a homomorphic image of R. Thus, by Lemma 3.2, the matrix A0 e1 0
is clean in M2n (Re1 ) (with In denoting the n × n identity matrix). Hence 0 In λe1 Ae1 is clean in Se1 . Similarly, e2 − λe2 ∈ Re2 is a central nilpotent and hence e2 − λe2 ∈ J(Re2 ). As Re2 I e −A e
0 is weakly clean, by Lemma 3.2 it follows that the matrix n 2 0 0 2 In (e2 −λe2 ) is clean in M2n (Re2 ), hence 0 0 In e2 − A0 e2 A0 e2 I2n e2 − = 0 In (e2 − λe2 ) 0 In λe2 is clean in M2n (Re2 ). Hence Ae2 is clean in Se2 .
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Lastly, since λe3 and e3 − λe3 are invertible in Re3 , by Lemma 3.1 the matrix A0 e3 0
is clean in M2n (Re3 ). Thus Ae3 is clean in Se3 . We conclude that each 0 In λe3 matrix Aei is clean in Sei , and hence A is clean in S. This completes the proof of the theorem. 2 Now we draw some consequences of the theorem. Corollary 3.6. Let R be an algebra over a field F . Then R is weakly clean if and only if it is a corner ring of a clean F -algebra. Proof. Apply Theorem 3.5 with k = F .
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We do not know if every unital C ∗ -algebra with real rank zero is clean. However, by Corollary 3.6 we have: Corollary 3.7. A complex unital C ∗ -algebra has real rank zero if and only if it is a corner ring of a clean C-algebra. 2 Corollary 3.8. Let R be a ring with char(R) = 0. Then R is weakly clean if and only if it is a corner ring of a clean ring. Proof. If char(R) = n = 0 then Z(R) contains a π-regular subring Zn = Z/nZ. Hence we may apply Theorem 3.5 with k = Zn . 2 The center of a π-regular ring is π-regular (see [10, Theorem 1]). Thus, by Theorem 3.5, we have: Corollary 3.9. Every π-regular ring is a corner of a clean ring.
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In light of the above results, it is natural to ask: Question 3.10. Is every weakly clean ring a corner ring of some clean ring? The affirmative answer to this question would give a characterization of weakly clean rings as the class of corners of clean rings. We conclude by noting that the center of a clean ring need not be clean (see [4, Proposition 2.5]). Thus the methods of Theorem 3.5 seemingly cannot be applied for arbitrary weakly clean rings. 4. Extensions of weakly clean rings Ara in [1] defined the notion of a non-unital exchange ring and proved the extension theorem for exchange rings. We will prove a similar result (with some more assumptions)
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for weakly clean rings. First, let us introduce the notion of a weakly clean non-unital ring. Definition 4.1. Let I be a non-unital ring and a ∈ I. Then a is called weakly clean in I if it can be written as a = e + p + exa for some e ∈ Id(I), x ∈ I, and p an element of the set Q(I) = {p ∈ I | there exists q ∈ I such that p + q + pq = p + q + qp = 0}. A ring I is weakly clean if each a ∈ I is weakly clean in I. Remark 4.2. (i) A simple verification shows that the above definition matches with the standard definition if I has a unity. Next, every non-unital clean ring (in the sense of Nicholson and Zhou [14]) is weakly clean, and every weakly clean non-unital ring is exchange (in the sense of Ara [1]). (ii) If I is a non-unital subring of a ring R such that I is weakly clean, then every a ∈ I is weakly clean (in the unital sense) in R. Indeed, if a ∈ I, then −a ∈ I and by assumption, −a = e + p + exa with e = e2 ∈ I, p ∈ Q(I) and x ∈ I. It follows that a = (1 − e) − (1 + p) − exa, with 1 − e ∈ Id(R) and 1 + p ∈ U (R). (iii) Conversely, if I is an ideal of a unital ring R such that every a ∈ I is weakly clean in R, then I is a weakly clean ring. Indeed, for any a ∈ I, −a is weakly clean in R, i.e. −a = e + u + f xa, with e = 1 − f ∈ Id(R), u ∈ U (R) and x ∈ R. We have e + u ∈ I, hence f u ∈ I and f ∈ I. Thus a = f − (1 + u) − f · f x · a, with −1 − u ∈ Q(I) and f x ∈ I. (iv) The above conclusion fails if I is only a subring (not an ideal). For example, take I = Z and R = Q; then every a ∈ I is clean in R, but I is not weakly clean. (v) The non-unital weakly clean property is left–right symmetric. The easiest way to prove this is through observation that an element a of a non-unital ring I is weakly clean in I if and only if (0, −a) is a weakly clean element of the unitization I 1 = Z ⊕ I of I. The left–right symmetry then follows from the unital left–right symmetry. Example 4.3. (1) A (possibly non-unital) C ∗ -algebra is weakly clean if and only if it has real rank zero. Indeed, if A is a weakly clean non-unital C ∗ -algebra, then by Remark 4.2 (i) A is exchange, hence by [1, Theorem 3.8] it has real rank zero. Conversely, if A has real rank zero, then, by definition, its unitization A1 = C ⊕ A has real rank zero (see [3]) and is therefore weakly clean by Example 2.7 (3). A is an ideal of A1 , hence by Remark 4.2 (iii), A is also weakly clean. (2) Every non-unital π-regular ring is weakly clean (this can be easily verified by adapting the proof of the unital case). Recall that if R is a (unital) ring, e ∈ R an idempotent and I an ideal of R, then eIe = {exe | x ∈ I} is an ideal of the corner ring eRe.
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Lemma 4.4. Let I be an ideal of a ring R and e ∈ Id(R). If I is weakly clean then eIe is weakly clean. Proof. Suppose that I is weakly clean and take a ∈ eIe. Since a is weakly clean in I, by Remark 4.2 (ii) it is also weakly clean in R and hence weakly clean in eRe. Thus every a ∈ eIe is weakly clean in eRe. By Remark 4.2 (iii) it follows that eIe is weakly clean. 2 Recall that by Example 2.7 (1), regular rings are weakly clean. The following theorem extends this result: Theorem 4.5. Let R be a ring and I an ideal such that I is weakly clean, R/I is regular and idempotents lift modulo I. Then R is weakly clean. Proof. In the proof we will use techniques similar to those in [1, Proof of Theorem 2.2]. Let R and I satisfy the assumptions of the theorem. For any x ∈ R, we denote by x the homomorphic image of x in R = R/I. Take any element a ∈ R, and denote α = a. Since R is regular, there exists β ∈ R such that αβα = α. Since αβ is an idempotent in R, and idempotents lift modulo I, there exists e ∈ Id(R) such that e = αβ. Write β = b for some b ∈ R. Observe that eabe − e = 0, and therefore eabe − e ∈ I ∩ eRe = eIe. Now, since eIe is weakly clean, there exist g ∈ Id(eIe), p ∈ Q(eIe) and x ∈ eIe such that eabe − e = g + p + gx(eabe − e). Writing u = e + p ∈ U (eRe), we have eabe = g + u + gx(eabe − e).
(1)
Let v denote the inverse of u in eRe, and h = e − g. Multiplying (1) by vh from the left, we have vhabe = vhu. It follows that (bvha)2 = bvhuvha = bvha, and hence q = bvha is an idempotent in Ra. By Lemma 2.6, it suffices to see that (1 − q)a(1 − q) is weakly clean in (1 − q)R(1 − q). Observe that u = e, hence v = e and q = βv(e − g)α = βeα = βα. Thus we have (1 − q)a(1 − q) = (1−βα)α(1−βα) = 0. We conclude that (1−q)a(1−q) ∈ (1 − q)R(1 − q) ∩ I = (1 − q)I(1 − q). Since (1 − q)I(1 − q) is weakly clean, the theorem is proved. 2 Theorem 4.5 cannot be generalized to the case when R/I is semiregular or even local: Example 4.6. There exists a ring R with an ideal I such that I is a clean non-unital ring, idempotents lift modulo I and R /I is local, but R is not weakly clean. Let I and R be as in [15, Example 3.1]. Following notation in [15], we denote by ψ : R → F ((X)) a surjective ring homomorphism with the kernel I, where F ((X)) denotes the field of formal Laurent series over the field F . Note that R is a weakly clean ring. Now, let k ⊆ F ((X)) be a subring of all rational functions p(X)q(X)−1 , with p(X), q(X) ∈ F [X] and q(X) = 0, such that q(X) is not divisible by X − 1. Observe
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that k is a local ring since it is isomorphic to the localization of the polynomial ring F [X] at the prime ideal F [X](X − 1). The Jacobson radical of this ring is precisely J(k) = k(X − 1). Now define
R = A ∈ R | ψ(A) ∈ k . The quotient mapping ψ : R → F ((X)) induces a surjection ψ|R : R → k with the kernel I. Clearly, idempotents in k lift modulo I. The ring I is clean (cf. [15, Proposition 4.2]). It remains to show that R is not weakly clean. Let A ∈ R denote the left shift operator, so that ψ(A) = X. Let us prove that 1 − A is not weakly clean in R . Suppose, to the contrary, that E = T (1 − A) and 1−E = (1−E)U A, with E ∈ Id(R ), T ∈ R and U ∈ U (R ). Since ψ(E) is an idempotent and ψ(E) = ψ(T )(1 − X) ∈ J(k), it follows that ψ(E) = 0. Hence ψ(1 − E) = 1, which gives 1 = ψ(U )ψ(A) and ψ(U ) = ψ(A)−1 = X −1 . But this contradicts the fact that U is invertible in R. Hence 1 − A is not weakly clean. Remark 4.7. (i) The ring in the above example is an exchange ring by [1] (I and R /I are exchange rings and idempotents lift modulo I). Thus we answer in negative the question in [15], asking whether every exchange ring is weakly clean. (ii) Note that the element A in the above example is weakly clean but 1 − A is not. Thus, this example also shows that the weakly clean property is not symmetric in the sense that if a is weakly clean then 1 − a is weakly clean. In Example 4.6, the fact that R is not weakly clean follows from the nonzero radical of the factor ring R /I. The following example shows that the nonzero radical of the factor ring is not crucial. Example 4.8. There exists a ring S with an ideal J such that J is clean, S/J is clean with J(S/J) = 0 and idempotents lift modulo J, but S is not weakly clean. Let I, R and R be as in Example 4.6. We repeat the construction performed in [12, Example 1.7]. Define
S = (a1 , . . . , an , a, a, . . .) | ai ∈ R and a ∈ R and
J = (a1 , . . . , an , a, a, . . .) | ai , a ∈ I . Then S is a ring with an ideal J such that
S/J ∼ = (t1 , . . . , tn , t, t, . . .) | ti ∈ F (X) and t ∈ k .
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Observe that J and S/J are clean rings, J(S/J) = 0 and idempotents lift modulo J, but S is not weakly clean since R is its homomorphic image. We do not know whether Theorem 4.5 can be extended to the case when R/I is π-regular or even artinian. Note that the factor rings R /I and S/J in the above examples are not π-regular. However, we are able to give the following partial answer to this question. Proposition 4.9. Let R be a ring with an ideal I such that I and R/I are both π-regular rings. Then R is weakly clean. Note that in this proposition we have no assumption that idempotents lift modulo I. Idempotent lifting indeed comes automatically: Lemma 4.10. (See [11, Lemma 3].) If R is any ring and I an ideal of R such that I is π-regular then idempotents lift modulo I. Proof of Proposition 4.9. Suppose that R is a ring and I an ideal such that I and R/I are π-regular rings. For any x ∈ R, we denote by x the homomorphic image of x in R = R/I. Take any element a ∈ R, and denote α = a. Since R is π-regular, there exist β ∈ R and n 1 such that αn βαn = αn . Since αn β is an idempotent in R, by Lemma 4.10 there exists e ∈ Id(R) such that e = αn β. Write β = b for some b ∈ R. Observe that ean be − e = 0 and therefore ean be − e ∈ I ∩ eRe = eIe. Since eIe is weakly clean, there exist g ∈ Id(eIe), p ∈ Q(eIe) and x ∈ eIe such that ean be − e = g + p + gx(ean be − e). Writing u = e + p ∈ U (eRe) we have
ean be = g + u + gx ean be − e . Let v denote the inverse of u in eRe, and h = e − g. Multiplying the above equation by vh from the left, we have vhan be = vhu. It follows that (bvhan )2 = bvhuvhan = bvhan , and hence q = bvhan is an idempotent in Ra. By Lemma 2.6, it suffices to prove that (1 − q)a(1 − q) is weakly clean in (1 − q)R(1 − q). Furthermore, by the note following Proposition 2.5 it suffices to prove that (1 − q)a(1 − q) is weakly clean in R. Observe that u = e, hence v = e and q = βv(e − g)αn = βeαn = βαn . As in Example 2.7 (1) we see that (α(1 − βαn ))n = 0. Hence (a(1 − q))n ∈ I and therefore ((1−q)a(1−q))n ∈ I. Since I is π-regular, it follows that ((1−q)a(1−q))n is π-regular in I and hence (1−q)a(1−q) ∈ R is π-regular in R. Hence by Example 2.7 (1) (1−q)a(1−q) is weakly clean in R. This completes the proof. 2 It is a natural question if the assumptions of Proposition 4.9 actually imply that the ring R is π-regular. In this direction we know only one partial result by Fuchs and Rangaswamy:
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Lemma 4.11. (See [8, Lemma 1].) Let R be a ring with an ideal I such that I is π-regular and for every a ∈ R there exist x ∈ R and n 1 with ax = xa and an xan − an ∈ I. Then R is π-regular. This result suggests that the answer in general is negative. Thus, Proposition 4.9 probably presents a non-trivial generalization of Example 2.7 (1). Acknowledgment I would like to thank Slovenian Research Agency for financial support. References [1] P. Ara, Extensions of exchange rings, J. Algebra 197 (2) (1997) 409–423. [2] P. Ara, K.R. Goodearl, K.C. O’Meara, E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998) 105–137. [3] L.G. Brown, G.K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1) (1991) 131–149. [4] W.D. Burgess, R. Raphael, On embedding rings in clean rings, Comm. Algebra 41 (2) (2013) 552–564. [5] V.P. Camillo, D. Khurana, A characterization of unit regular rings, Comm. Algebra 29 (5) (2001) 2293–2295. [6] V.P. Camillo, D. Khurana, T.Y. Lam, W.K. Nicholson, Y. Zhou, Continuous modules are clean, J. Algebra 304 (1) (2006) 94–111. [7] V.P. Camillo, H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (12) (1994) 4737–4749. [8] L. Fuchs, K.M. Rangaswamy, On generalized regular rings, Math. Z. 107 (1968) 71–81. [9] K.R. Goodearl, R.B. Warfield Jr., Algebras over zero-dimensional rings, Math. Ann. 223 (2) (1976) 157–168. [10] N.H. McCoy, Generalized regular rings, Bull. Amer. Math. Soc. 45 (2) (1939) 175–178. [11] P. Menal, On π-regular rings whose primitive factor rings are Artinian, J. Pure Appl. Algebra 20 (1) (1981) 71–78. [12] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278. [13] W.K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra 27 (8) (1999) 3583–3592. [14] W.K. Nicholson, Y. Zhou, Clean general rings, J. Algebra 291 (1) (2005) 297–311. [15] J. Šter, Corner rings of a clean ring need not be clean, Comm. Algebra 40 (5) (2012) 1595–1604. [16] J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (2) (1986) 437–453.