Torsion functors, small or large

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.