The group invertibility of matrices over Bézout domains

Abstract

Let R be a Bézout domain, and let A,B,C ∈ Rn×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, Electronic J. Linear Algebra, 18 (2009), 600–612).