A note on semicentral Idempotents

TL;DR

This paper characterizes rings where idempotents isomorphic to semicentral idempotents are also semicentral, showing these are exactly Dedekind-finite rings, and explores implications in module theory.

Contribution

It provides a complete characterization of rings with this property, linking semicentral idempotents to Dedekind-finiteness, and extends the discussion to module theory.

Findings

Rings with isomorphic semicentral idempotents are Dedekind-finite.

Dedekind-finite rings are precisely those where the property holds.

Application to module theory demonstrates broader implications.

Abstract

In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.