The group invertibility of matrices over Bézout domains

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).

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