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We will give some glimpses into noncommutative ring theory by discussing some typical results. One of the aim is to demonstrate some basic ways in which noncommutative ring theory is different from its commutative version. [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]

It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules. [ABSTRACT FROM AUTHOR]

We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring R, then for any element a ∈ R and central units u1, u2, ..., un ∈ U(R) there exists a unit u ∈ U(R) such that a + uiu ∈ U(R) for each i ≥ 1. [ABSTRACT FROM AUTHOR]

In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring. [ABSTRACT FROM PUBLISHER]

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units. [ABSTRACT FROM AUTHOR]

Abstract: A ring S is said to have square stable range one (written ) if implies that is a unit for some . In the commutative case, this extends the class of rings of stable range 1, and allows many new examples such as rings of real-valued continuous functions, and real holomorphy rings. On the other hand, sometimes forces S to have stable range 1. For instance, this is the case for exchange rings S, for which is characterized by being reduced (or abelian, or right quasi-duo). We also characterize rings S whose (von Neumann) regular elements are strongly regular, by using an element-wise notion of square stable range one. Extending a result of Estes and Ohm, we show that a possibly noncommutative infinite domain with stable range one or square stable range one must have a non-artinian group of units. [Copyright &y& Elsevier]

A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over ℤ2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to ℤ2. [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]

In this paper, we study the modules whose any two isomorphic summands have a common complement. We call such modules perspective. It is proved that perspectivity is an ER-property, in the sense that it depends only on the endomorphism ring of the module. This property in rings turns out to be left-right symmetric, that is, RR is perspective if and only if RR is perspective for any ring R and we call such ring a perspective ring. The class of perspective rings lies properly between the class of rings with stable range one and the class of rings with the internal cancellation property. We give several characterizations of perspective rings and also study perspectivity in related rings. It is proved that R has stable range one if and only if M2(R) is a perspective ring. Another characterization of rings with stable range one, in terms of completions of unimodular rows, is also given. We prove that every regular element in a perspective ring is clean. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if is commutative. Many other characterizations are obtained for such rings R in terms of (the group of units of R), including, for instance, , and (for all ). We also consider a natural weakening of these conditions, namely, , and show that, for von Neumann regular rings, this is a (necessary and) sufficient condition for the commutativity of R. [Copyright &y& Elsevier]

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]