The group invertibility of matrices over B\'ezout domains

Let $R$ be a B'ezout 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 $(AB{)}^{\mathrm{\#}}$ is similar to $(CA{)}^{\mathrm{\#}}$. This generalize the main result of Cao and Li(Group inverses for matrices over a B'ezout domain, {\it Electronic J. Linear Algebra}, {\bf 18}(2009), 600–612).