Torsion functors, small or large

TL;DR

This paper compares small and large torsion functors in commutative algebra, highlighting their differences in non-noetherian rings and exploring their properties and relationships.

Contribution

It introduces and analyzes the properties of small and large torsion functors, emphasizing their divergence in non-noetherian contexts and providing illustrative examples.

Findings

Small and large torsion functors coincide in noetherian rings.

Several properties of torsion functors do not extend to non-noetherian rings.

The paper presents examples demonstrating these differences.

Abstract

Let $\mathfrak{a}$ be an ideal in a commutative ring $R$. For an $R$-module $M$, we consider the small $\mathfrak{a}$-torsion $\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^n\subseteq(0:_Rx)\}$ and the large $\mathfrak{a}$-torsion $\overline{\Gamma}_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:_Rx)}\}$. This gives rise to two functors $\Gamma_{\mathfrak{a}}$ and $\overline{\Gamma}_{\mathfrak{a}}$ that coincide if $R$ is noetherian, but not in general. In this article, basic properties of as well as the relation between these two functors are studied, and several examples are presented, showing that some well-known properties of torsion functors over noetherian rings do not generalise to non-noetherian rings.