Commutative rings over which the support of any module is the collection of prime ideals containing the annihilator

TL;DR

This paper characterizes commutative rings where the support of every module equals the set of prime ideals containing its annihilator, providing a complete classification and extending the result to countably generated modules.

Contribution

It completely classifies commutative rings with this support property and shows it suffices for countably generated modules.

Findings

Identifies rings where support equals prime ideals containing annihilators for all modules.

Extends the classification to countably generated modules.

Provides a complete characterization of such rings.

Abstract

The support of any module over a commutative ring is defined as the collection of all prime ideals of the ring at which the localization of the module is non-zero. For finitely generated modules, the support is the collection of all prime ideals containing the annihilator of the module. In this article, we raise the natural question that over which commutative rings, the support of every module is the collection of all the prime ideals of its annihilator. We completely classify such rings, and in the process it also comes out that it is enough to require that only for countably generated modules, the support is the collection of all prime ideals containing the annihilator of the module.