On Exchange QB-Rings.

In this paper, we establish necessary and sufficient conditions for an exchange ring R to be a QB-ring. We show that an exchange ring R is a QB-ring if and only if for any regular x ∈ R, there exists u ∈ such that ux ∈ R is an idempotent, if and only if whenever eR ≅ fR with idempotents e, f ∈ R, there exists u ∈ such that eu = uf. As an application, we also prove that every square matrix over exchange QB-rings admits a diagonal reduction by some quasi-inverse matrices. These give generalizations of many known results such as Theorem 2A by Hartwing and Luh (1977). A note on the group structure of unit regular ring elements. Pacific J. Math. 71:449–461.) and Theorem by Guralnick and Lanski (Guralnick, R., Lanski, C. (1982). Pseudosimilarity and cancellation of modules. Linear Algeb. Appl. 47:111–115.). [ABSTRACT FROM AUTHOR]