Generalized Stable Exchange Rings

Abstract

A ring R is said to be a generalized stable ring provided that aR + bR = R with a, b ∈ R implies that there exists y ∈ R such that a + by ∈ K(R), where K(R) = {x ∈ R | ∃s, t ∈ R such that sxt = 1}. Let A be a quasi-projective right R-module, and let E = End R(A). If E is an exchange ring, then E is a generalized stable ring if and only if for any R-morphism f : A → M with Im f ≤⊕M and any R-epimorphism g : A → M, there exist e = e2 ∈ E and h ∈ K(E) such that f = g(eh). Furthermore, we prove that every regular matrix over a generalized stable exchange ring admits a diagonal reduction by quasi-invertible matrices.