Stable range one for exchange rings

Abstract

An associative ring R with identity is called an exchange ring if R R has the exchange property introduced by Crawley and Jonsson (1964). We prove, in this paper, that an exchange ring R has stable range 1 in the following three cases: (1) primitive factors of R are Artinian; (2) R is of bounded index of nilpotence; (3) idempotents of R are central. Also it is shown that, for exchange rings, bounded index of nilpotence implies primitive factors are Artinian and that, for modules with the finite exchange property, cancellation from direct sums is equivalent to the fact that the endomorphism ring of the cancellable module has stable range 1.