«CLASSES OF ALMOST CLEAN RINGS ˇ EVRIM AKALAN AND LIA VAS Abstract. A ring is clean (almost clean) if each of its elements is the sum of a unit ...»
CLASSES OF ALMOST CLEAN RINGS
EVRIM AKALAN AND LIA VAS
Abstract. A ring is clean (almost clean) if each of its elements is the sum of a unit
(regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular
module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including ﬁnite AW ∗ -algebras and noetherian Leavitt path algebras in particular, are almost clean.
We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart.
If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasicontinuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean.
Finally, we adapt some of our results to rings with involution: a ∗-ring is ∗-clean (almost ∗-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) ∗-clean ring is similarly deﬁned by replacing “idempotent” with “projection” in the appropriate deﬁnition. We show that an abelian ∗-ring is a Rickart ∗-ring if and only if it is special almost ∗-clean, and that an abelian ∗-ring is ∗-regular if and only if it is special ∗-clean.
Introduction A ring is clean if each of its elements can be written as the sum of a unit and an idempotent. W. K. Nicholson introduced the concept of clean rings in the late 1970s.
Since then, some stronger concepts (e.g. uniquely clean, strongly clean, and special clean rings) have been considered, as well as some weaker ones (e.g. almost clean rings).
2000 Mathematics Subject Classiﬁcation. 16U99, 16W99, 16W10, 16S99.
Key words and phrases. Clean, almost clean, quasi-continuous, nonsingular, Rickart, abelian and CS rings.
This work was supported by the Visiting Scientists Fellowship Program grant from The Scientiﬁc and Technological Research Council of Turkey (TUBITAK) and was carried out in part during the visit of the second author to the Hacettepe University in July 2011. The second author is grateful to the faculty and staﬀ of Hacettepe University for their hospitality and support. The authors are also grateful to all who helped improve the language of the paper.
2 EVRIM AKALAN AND LIA VASA ring is almost clean if each of its elements can be written as the sum of a regular element (neither a left nor a right zero-divisor) and an idempotent. Almost clean rings were introduced in  for commutative rings where it is shown that a commutative Rickart ring is almost clean. In most papers so far, almost cleanness is considered in the commutative case. One of the exceptions is  which proves that certain Baer ∗-rings (in particular ﬁnite AW ∗ -algebras of type I) that are not necessarily commutative are almost clean. This result was shown by embedding such a Baer ∗-ring R in the maximal right ring of quotients Qr (R). The ring Qr (R) is unit-regular (thus clean) and has max max the same projections (self-adjoint idempotents) as R. In this situation, the cleanness of Qr (R) implies that the ring R is almost clean.
max In this paper, we extend and generalize the idea of : we consider a class of rings R with the property that R embeds in a clean ring with the same idempotents as R. All right quasi-continuous and right nonsingular rings have this property. We show that they are almost clean (Proposition 2.3). We also generalize our results to modules (Theorem 2.6).
In Theorem 2.7, we show that the assumption that R is right quasi-continuous (C1+C3) can be relaxed to the assumption that R is right CS (right extending, i.e. (C1)).
As a corollary, we also show that the class of right strongly semihereditary rings, studied in , is almost clean (Corollary 2.10). Consequently, all ﬁnite AW ∗ -algebras are almost clean. This fact extends the results in : we now know that all ﬁnite AW ∗ -algebras are almost clean, not just ﬁnite AW ∗ -algebras of type I. In part, this result contributes to determining those von Neumann algebras that are clean (an initiative started by T.Y.
Lam). In addition, our result also implies that all noetherian Leavitt path algebras (Leavitt path algebras over ﬁnite no-exit graphs) are almost clean.
Clean rings are an additive analogue of unit-regular rings. In a unit-regular ring, each element can be written as the product of a unit and an idempotent. In the case of clean rings, “the product” in the last condition changes to “the sum”. The Camillo-Khurana Theorem in  characterizes unit-regular rings as clean rings in which each element a has the form a = u + e where u is a unit and e is an idempotent with aR ∩ eR = 0. Following the terminology used in , we refer to the rings satisfying the last condition as special clean rings.
Our goal is to establish a result analogous to the Camillo-Khurana Theorem: the exact relation between abelian Rickart rings, the rings in which each element can be written as the product of a regular element and an idempotent, and their additive analogues, almost clean rings. We show that an abelian ring is Rickart if and only if each element a has the form a = u + e where u is a regular element and e is an idempotent with aR ∩ eR = 0 (Theorem 3.1). We refer to the rings satisfying the last condition as special almost clean rings. Interestingly, this result has a corollary that a right quasi-continuous and right nonsingular ring is both left and right Rickart (Corollary 3.4). Note that [3, Theorem 3.2] demonstrates that a right quasi-continuous and right nonsingular ring R is right Rickart.
In this situation, our result shows that R is left Rickart as well.
We show that in the abelian case, the Camillo-Khurana Theorem can be strengthened to state that R is unit-regular if and only if it is uniquely special clean, i.e. special
CLASSES OF ALMOST CLEAN RINGS 3clean decompositions are unique (Proposition 4.1). As a corollary, we deduce that all abelian, right quasi-continuous, right nonsingular rings are uniquely special almost clean, i.e. special almost clean decompositions are unique (Corollary 4.2). Furthermore, an abelian, right quasi-continuous ring is Rickart if and only if it is uniquely special almost clean (Corollary 4.3).
Finally, we turn to ∗-rings and study their cleanness in the context of the presence of an involution. In , a ring with involution is said to be ∗-clean (almost ∗-clean) if each of its elements is the sum of a unit (regular element) and a projection. We deﬁne special ∗clean and special almost ∗-clean rings by replacing “idempotent” with “projection” in the deﬁnitions of special clean and special almost clean, respectively. We show the ∗-version of our characterization of abelian Rickart rings: an abelian ∗-ring is a Rickart ∗-ring if and only if it is special almost ∗-clean (Theorem 5.2). We also show the ∗-version of the Camillo-Khurana Theorem in the abelian case: an abelian ∗-ring is ∗-regular if and only if it is special ∗-clean (Theorem 5.3).
The paper is organized as follows. In Section 1, we recall some known concepts and results. In Section 2, we prove the results related to the almost cleanness of quasi-continuous or CS rings and modules. In Section 3, we prove a theorem on abelian Rickart rings analogous to the Camillo-Khurana Theorem and derive several related corollaries. In Section 4, we study the uniqueness of special clean and almost special clean decompositions. In Section 5, we adapt our earlier results to rings with involution. We conclude the paper with a list of open problems in Section 6.
As (C2) implies (C3), a continuous module is quasi-continuous ([12, Exercise 36, p.
245]). Also, if R is regular, then the following are equivalent: R is continuous, R is quasi-continuous and R is CS ( [12, Exercise 36, p. 246]).
The following result of Goel and Jain from  can also be found in [12, Exercise 37, p.
Proposition 1.1.  For any module M, the following are equivalent.
(1) M is quasi-continuous, (2) Any idempotent endomorphism of a submodule of M extends to an idempotent endomorphism of M.
(3) M is invariant under any idempotent endomorphism of the injective envelope E(M ).
Now, let us turn to the preliminaries on clean rings. A ring element is right (left) regular if it does not have nontrivial right (left) annihilators. It is regular if it is left and right regular. Note that we use the term “regular” as in, for example,  and not in the sense that a ∈ R is regular if a = axa for some x ∈ R.
We have reviewed the deﬁnitions of (almost) clean, special (almost) clean, and uniquely special (almost) clean rings in the introduction. Using the term “special clean”, the Camillo-Khurana Theorem can be stated as follows.
[5, Theorem 1] A ring R is unit-regular if and only if it is special clean.
Recall that an R-module M is called (almost) clean if its endomorphism ring is (almost) clean. In [6, Theorem 3.9] one ﬁnds the deep result that a continuous module is clean.
Finally, we note the following.
Proposition 1.3. [6, Proposition 4.5] If M is CS, then each endomorphism of M is the sum of an idempotent and a monomorphism.
Since an injective endomorphism of M is a right regular element of the endomorphism ring End(M ), we have the following implications for an arbitrary element f of End(M ).
Proof. Let Q denote a clean ring with the same idempotents as R in which R embeds.
Let a be an element in R. Then a is in Q as well. Thus, a = u + e for some idempotent e ∈ Q and unit u ∈ Q. By assumption, e is in R. Thus, u = a − e is in R as well. Since u is a unit in Q, 0 = annQ (u) ⊇ annR (u) and the same holds for the left annihilators. Thus, r r u is regular.
The last sentence of the proposition follows since each regular element of a von Neumann regular ring is a unit.
We note that Proposition 2.1 is not valid if “almost” is deleted since Z can be embedded in Q with the same idempotents (0 and 1) and Z is not clean.
Proposition 2.3. If R is right quasi-continuous and right nonsingular, then R is almost clean.
Proof. If R is right nonsingular, Qr (R) is regular and right self-injective. Thus Qr (R) max max is clean by [6, Corollary 3.12]. Then R is almost clean by Proposition 2.1.
In the case when Qr (R) of a right quasi-continuous ring R is unit-regular, a stronger max conclusion than Proposition 2.3 holds as the following corollary shows.
Corollary 2.4. If R is right quasi-continuous and Qr (R) is unit-regular, then R is max special almost clean.
Proof. Since Qr (R) is unit-regular, it is special clean by Theorem 1.2. Let a = u + e max be a special clean decomposition in Qr (R) of an element a of R. Thus, aQr (R) ∩ max max eQr (R) = 0. Then aR ∩ eR = 0 as well. The idempotent e is in R because R is quasimax continuous. Thus, u has to be in R as well and it has to be regular just like in the proof of Proposition 2.1. Thus, a = u + e is a special almost clean decomposition of a in R.
Next we consider quasi-continuous modules and prove a stronger version of Proposition 2.3 – we prove that it holds for modules as well. First, we show a preliminary proposition.
Recall that an endomorphism of a module M is said to be essential if its image is an essential submodule of M.
Proposition 2.5. If M is quasi-continuous, then every endomorphism of M is the sum of an idempotent and an essential monomorphism.
Proof. Let f be an endomorphism of M. Then f can be extended to the injective envelope E(M ) of M. Let f denote this extension.
The module E(M ) is injective and, therefore clean by [6, Corollary 3.11]. There exist an idempotent e and a unit u in the ring of endomorphism of E(M ) such that f = e + u.
The restriction e of the idempotent e to M is in End(M ) since M is quasi-continuous.
If u denotes the restriction of u to M, it is clearly a monomorphism. We claim that the image u(M ) is essential in M.
First, note that u is M -invariant since u = f − e. Then, note that the fact that M being essential in E(M ) implies that u(M ) is essential in u(E(M )) since u is a monomorphism.
In addition, u is a unit and so it is onto. Thus, u(E(M )) = E(M ). This shows that u(M ) is essential in E(M ) and therefore essential in M as well.
6 EVRIM AKALAN AND LIA VASTheorem 2.6. If M is a quasi-continuous and nonsingular module, then M is almost clean.
Proof. Let M be a quasi-continuous, nonsingular module and f be an endomorphism of M. Using Proposition 2.5, f can be written as e + u where e is an idempotent and u is an essential monomorphism. Hence, u is right regular in End(M ). We need to prove that u is left regular as well. Let us assume that gu = 0 for some endomorphism g of M.
Then, the kernel of g contains the image u(M ) of u. Since u(M ) is essential in M, ker g is essential in M. Therefore, the module M/ ker g is singular ([12, Example 7.6 (3)]). The map g factors to a monomorphism from the singular module M/ ker g to the nonsingular module M. Hence, this map has to be zero ([12, Exercise 4, p. 269]). Then g is zero as well.
We can represent these results as additions to the diagram from the previous section as follows. In the diagram below, the arrows indicate implications. The ﬁrst column refers to the properties of an R-module M and the second two refer to the properties of an arbitrary element f of End(M ).