The group invertibility of matrices over Bézout domains

Abstract

Let $R$ be a Bézout domain, and let $A,B,C\in R^{n\times n}$ with $ABA=ACA$. If $AB$ and $CA$ are group invertible, we prove that $AB$ is similar to $CA$. Moreover, we have $(AB{)}^{\#}$ is similar to $(CA{)}^{\#}$. This generalize the main result of Cao and Li(Group inverses for matrices over a Bézout domain, {\it Electronic J. Linear Algebra}, {\bf 18}(2009), 600--612).

arXiv comment11 pages