On the stable range condition of exchange rings

Abstract

The purpose of this paper is to prove the following facts:An exchange ring R has stable range at most n if and only if for any module isomorphism Rⁿ ⊕ A ≅ R⊕B, A is isomorphic to a direct summand of B, if and only if for any regular element x of Rⁿ, x = xux for some unimodular column u of ⁿR. For any module M with the finite exchange property, the stable range of End_R(M) is at most n if and only if M satisfies the n-weak cancellation. These results generalize a result of Menal and Moncasi [5] and a recent result of Camillo and Yu [2].