Local Comparability of Exchange Ideals

Abstract

An exchange ideal I of a ring R is locally comparable if for every regular x ∈ I there exists a right or left invertible u ∈ 1 + I such that x = xux.We prove that every matrix extension of an exchange locally comparable ideal is locally comparable. We thereby prove that every square regular matrix over such ideal admits a diagonal reduction. (2010): 16S34, 16E50, 16U99, 13B99 Mathematics Subject Classification