Stable Range One for Rings with many Idempotents

Abstract

Abstract.An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R , there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly w-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menai [5].Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b £ R satisfying aR + bR = R, there exists y £ R such that a + by is a unit. This definition is left-right symmetric by Vaserstein [9, Theorem 2]. Furthermore, by a theorem of Kaplansky, all one-sided units are two-sided in rings having stable range 1 (cf. Vaserstein [10, Theorem 2.6]). It is well known that a (von Neumann) regular ring P has stable range 1 if and only if R is unit-regular (see, for example, Goodearl [4, Proposition 4.12]).Call a ring P strongly n-regular if for every element a £ R there exist a number n (depending on a) and an element x £ R such that a" -an+lx. This is in fact a two-sided condition [3]. It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menai [5] proved that strongly ^-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278).In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7t-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly ^-regular ring P has stable range 1 if powers of every regular element are regular. The latter is a generalization of the abovementioned result of Goodearl and Menai for strongly 7r-regular regular rings. As one can see from our proofs, rings in these classes have a large supply of idempotents.Throughout, R stands for an associative ring with identity and J(R) for the Jacobson radical of R . Modules are unitary right P-modules except otherwise specified. For other undefined terms, readers are referred to [4].Let Mr be a right P-module. Following Crawley and Jonsson [2], Mr is said to have the exchange property if for every module Ar and any two

Rings of square stable range one

“…In [CY,Theorem 3], Camillo and Yu proved that an exchange ring S has sr(S) = 1 iff S is an IC ring (that is, reg(S) = ureg(S)). In the following, we obtain a complete analogue of this result for the square stable range -by proving that S has ssr(S) = 1 iff S is a strongly IC ring (that is, reg(S) = sreg(S)).…”

Rings with internal cancellation

“…For instance, if G is a locally finite group, we can reduce the considerations to the case where G is finite. In that case, all matrix rings over A are artinian, so A is stably IC by 2.1 (5). If G is an abelian group, we may again assume that it is finitely generated, and an easy argument enables us to replace A by a Laurent polynomial ring B in finitely many commuting variables over a field.…”

“…As a by-product of our discussions, we give a uniform treatment for some of the main criteria for exchange rings to have stable range 1 obtained in [5,46], and [7][8][9]. The main advantage of our approach is that, after we have developed the basic properties of IC rings, various criteria for exchange rings to have stable range 1 are now automatic consequences.…”

On strongly Π-regular ideals

“…In recent years, Many authors have studied strongly -regular rings from di erent view points such as [2,[4][5][6][7][8][9][10][11]15,19,20]. In this paper, we extend strongly -regular rings and introduce the concept of strongly -regular ideal of a ring.…”