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

TL;DR

This paper addresses when FI-extending property transfers from the right regular module $R_R$ to finitely generated projective modules. Assuming ACC on right annihilators in $R$, it proves that $R_R$ is FI-extending if and only if every finitely generated projective right $R$-module is FI-extending. The argument hinges on decomposing fg projectives as direct summands of $R^t$, exploiting endomorphism rings and idempotent decompositions, and establishing nilpotency of certain relations to produce essential direct summands. The result delivers an affirmative answer to a question of Birkenmeier–Park–Rizvi, providing a precise criterion for FI-extending behavior in rings with ACC on right annihilators and clarifying how the FI-extending property propagates to fg projectives. This contributes to understanding how invariance and essentiality interact with direct-sum decompositions in the module category over such 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}).

Theorems & Definitions (7)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof