The Generalized Flanders' Theorem in Unit-regular Rings

TL;DR

This paper generalizes Flanders' theorem to unit-regular rings and complex matrices, establishing conditions under which products are similar or have similar Drazin inverses, extending prior results in ring and matrix theory.

Contribution

It extends Flanders' theorem to unit-regular rings and matrix products, providing new conditions for similarity and invertibility equivalences.

Findings

AC and BA are similar if and only if their powers have the same rank for matrices.

AC and BA are similar in unit-regular rings when certain invertibility conditions are met.

Drazin inverses of AC and BA are similar under specified conditions.

Abstract

Let R be a unit-regular ring, and let a,b,c in R satisfy aba=aca. If ac and ba are group invertible, we prove that ac is similar to ba. Furthermore, if ac and ba are Drazin invertible, then their Drazin inverses are similar. For any n\times n complex matrices A,B,C with ABA = ACA ,we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders' theorem proved by Hartwig.