Rings for Which f.g. Projective Modules Have the FI-extending Property

TL;DR

This paper characterizes rings with ACC on right annihilators where the regular module is FI-extending, showing this property is equivalent to all finitely generated projective modules being FI-extending, answering a question from 2002.

Contribution

It provides a complete characterization of rings with ACC on right annihilators for which the FI-extending property of the regular module implies the same for all finitely generated projective modules.

Findings

Rings with ACC on right annihilators have the FI-extending property for the regular module if and only if all finitely generated projective modules are FI-extending.

The paper confirms a question posed by Birkenmeier, Park, and Rizvi in 2002.

Establishes a clear equivalence condition for FI-extending modules over certain rings.

Abstract

A right $R$-module $M$ is said to be {\it FI-extending} if any fully invariant submodule of $M$ is essential in a direct summand of $M$. In this short note we prove that if $R$ has ACC on the right annihilators, then $R_R$ is FI-extending if, and only if, every f.g. projective module is too FI-extending. This is an affirmative answer to the question raised by Birkenmeier-Park-Rizvi in Commun. Algebra on 2002 (see \cite{2}).