Modules with ascending chain condition on annihilators and Goldie modules

TL;DR

This paper investigates modules with ascending chain condition on annihilators, establishing finiteness of minimal primes, characterizing indecomposable injectives, and providing new insights into Goldie modules and rings.

Contribution

It introduces new conditions under which modules have finitely many minimal primes and characterizes Goldie modules using injective decompositions.

Findings

Modules with ACC on annihilators have finitely many minimal primes.

Bijective correspondence between indecomposable injectives and minimal primes.

Goldie modules decompose into direct sums of uniform injective modules.

Abstract

Using the concepts of prime module, semiprime module and the concept of ascending chain condition (ACC) on annihilators for an $R$-module $M$ . We prove that if \ $M$ is semiprime \ and projective in $\sigma \left[ M\right] $, such that $M$ satisfies ACC on annihilators, then $M$ has finitely many minimal prime submodules. Moreover if each submodule $N\subseteq M$ contains a uniform submodule, we prove that there is a bijective correspondence between a complete set of representatives of isomorphism classes of indecomposable non $M$-singular injective modules in $\sigma \left[ M\right] $ and the set of minimal primes in $M$. If $M$ is Goldie module then $% \hat{M}\cong E_{1}^{k_{1}}\oplus E_{2}^{k_{2}}\oplus ...\oplus E_{n}^{k_{n}}$ where each $E_{i}$ is a uniform $M$-injective module. As an application, new characterizations of left Goldie rings are obtained.