Exchange rings having stable range one

Abstract

Abstract. We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979Goodearl ( , 1991, Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). We call R has stable range one provided that aR+bR = R implies that a+by ∈ U(R) for a y ∈ R. It is well known that an exchange ring R has stable range one if and only if A ⊕ B A ⊕ C implies B C for all finitely generated projective right R-modules A, B, and C. It has been realized that the class of rings having stable range one has good stability properties in a K-theoretic sense (cf. [8,13]). Many authors have studied stable range one conditions over exchange rings such as [1,2,4,5,6,7,15,16].In this paper, we investigate stable range one conditions over exchange rings by virtue of Drazin inverses, nilpotent elements, and prime ideals. We showed that stable range conditions can be determined by Drazin inverses for exchange rings. Also, we see that these stable range conditions can be determined by regular elements out of any proper ideal of R. Moreover, we prove that an exchange ring R has stable range one if the set of nilpotents is closed under product. These extend the corresponding results of [4, Theorem 3], [9, Theorem 4.19], [12, Theorem 2],and [16, Theorem 9].Throughout this paper, all rings are associative ring with identities and all right R-modules are unitary right R-modules. M ⊕ N means that right R-module M is isomorphic to a direct summand of right R-module N. The notation x ≈ y means that x = uyu −1 for some u ∈ U(R), where U(R) denotes the set of all units of R. Call a ∈ R is regular if a = axa for some x ∈ R and a ∈ R is unit-regular if a = aua for some u ∈ U(R).

Rings with internal cancellation

“…So by (4), x ∈ ureg(R), and thus x ∈ ureg(R). 2 Specializing 9.7 to exchange rings, we can now retrieve the following result of Chen [8,Theorem 2.2]. (In Chen's result, however, the crucial hypothesis that J = R was left out.)…”

“…This result is inspired by Chen's Theorem 2.2 in [8], which states that, if J is a proper ideal in an exchange ring R, then R has stable range 1 iff every regular element in R \ J is unit-regular. Here, we work more generally with the characterization of IC rings; Chen's theorem can be recaptured by simply specializing our result to exchange rings: see Corollary 9.8.…”

“…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.…”

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

“…In generalization of (5.5) ((1) ⇔ (2)), Camillo and Yu [13] have shown that, for any exchange ring R, sr(R) = 1 iff every regular element of R is unit-regular, and H. Chen [17] showed that sr(R) = 1 is also equivalent to aR = bR ⇒ a ∈ b · U(R) (for a, b ∈ R). For other characterizations, see [88], [17,18,20], and [54]. More recently, the case of stable range n has been studied over exchange rings; see, for instance, [17], [22,23], [21], and [87]; see also items (J) and (K) below.…”