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We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam. [ABSTRACT FROM AUTHOR]

We investigate connections between association chains of idempotents and bounded generation of SL 2 by elementary matrices. This enables us to identify many classes of matrix rings that have universal bounds on the length of association chains. In particular, many matrix rings have the property that every regular element is special clean. We also improve the usual criterion for checking perspectivity via association chains. [ABSTRACT FROM AUTHOR]

A clean decomposition a = e + u in a ring R (with idempotent e and unit u) is said to be special if a R ∩ e R = 0. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution. [ABSTRACT FROM AUTHOR]

We study rings in which ab nonzero idempotent implies ba is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings. [ABSTRACT FROM AUTHOR]

It is well-known that idempotents lift modulo any nil one-sided ideal. While this is not true for periodic elements, it does hold true in many special cases. We investigate the connections between these special cases, as well as limitations. We also answer three questions from the literature. For example, we construct a nilpotent ideal where torsion-units lift, but periodic elements do not lift, modulo that ideal. [ABSTRACT FROM AUTHOR]

In analogy to the fact that isomorphic idempotents arise from flipped factorizations, we show that under some weak conditions, partial factorizations of the complement idempotents may also be preserved. Applying these results, we give new characterizations of exchange rings. We also show that the definition of an exchange element is left–right symmetric. [ABSTRACT FROM AUTHOR]

The equation XA-BX = I has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where B2 = B, the study of XA - BX = I in a noncommutative ring R leads to several new ways to view and to work with the exchange (or "suitable") elements in R in the sense of Nicholson. For any exchange element A ∊ R, we show that the set of idempotents E ∊ R such that E ∊ RA and I - E ∊ R(I - A) is naturally parametrized by the roots of a certain left-right symmetric "exchange polynomial" associated with A. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood. [ABSTRACT FROM AUTHOR]

In this paper, we study exchange rings and clean rings with (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic . [ABSTRACT FROM AUTHOR]

Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu-1 for some u ∈ U(R), if and only if ef ≠ 0. [ABSTRACT FROM AUTHOR]

It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n. We prove that if n is a unit in R , then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I. By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R , we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I. It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π -regular ring. [ABSTRACT FROM AUTHOR]

It is proved that a ring R is unit regular if and only if every element a of R can be written as e + u such that aR ∩ eR = 0, where e is an idempotent and u a unit in R. [ABSTRACT FROM AUTHOR]

Abstract: Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if is a ring which is complete with respect to an ideal and if is an element of whose image in is strongly -regular, then is strongly clean in . This generalizes Theorem 2.1 of Chen and Zhou (2007) [9]. [Copyright &y& Elsevier]