Search

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]

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]

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 ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 -a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a1, a2) ∈ R2, one of the ai's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring Tn(R) is VNL if and only if n = 2 or 3 and R is a division ring. [ABSTRACT FROM AUTHOR]