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 ${\Gamma }_{\mathfrak{a}}(M)=\{x\in M\mid \exists n\in \mathbb{N}:{\mathfrak{a}}^n\subseteq (0{:}_{R}x)\}$ and the large $\mathfrak{a}$-torsion ${\overline{\Gamma }}_{\mathfrak{a}}(M)=\{x\in M\mid \mathfrak{a}\subseteq \sqrt{(0{:}_{R}x)}\}$. 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.

To appear in Beitr. Algebra Geom.; some typos corrected

References in corpus (2)

• Injective modules and torsion functors
• Assassins and torsion functors

Cited by in corpus (3)

• The locally nilradical for modules over commutative rings
• Generalised reduced modules
• Assassins and torsion functors II