Torsion functors, small or large

Abstract

Let a be an ideal in a commutative ring R. For an R-module M , we consider the small a-torsion Γ a (M ) = {x ∈ M | ∃n ∈ AE : a n ⊆ (0 : R x)} and the large a-torsion Γ a (M ) = {x ∈ M | a ⊆ (0 : R x)}. This gives rise to two functors Γ a and Γ 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 nonnoetherian rings.

Assassins and Torsion Functors

“…6, 3.2]). The functors Γ a and Γ a are called the small a-torsion functor and the large a-torsion functor; they need not be equal ([6,Section 4]). 6, 3.4, 3.5]).…”

“…We collect basic facts about torsion functors, assassins, and weak assassins. For details we refer the reader to [6], [2, Chapter IV] (especially Exercice IV.1.17) and [8, 00L9, 0546]. 6, 3.2]).…”

“…If R is noetherian, the two subsets of Spec(R) coincide, but they need not do so in general. In [6], the functors Γ a and Γ a were investigated extensively. In [5], the relations between assassins and weak assassins of Γ a (M), M and M/Γ a (M) were studied.…”

“…a) If a is idempotent and nonzero, it is neither half-centred nor centred. b) If Γ a is a radical, a is fair 6. Proof.…”

The locally nilradical for modules over commutative rings

“…where Γ a (M) is the submodule of M given by Γ a (M) := m ∈ M | a k m = 0 for some k ∈ Z + . On modules defined over Noetherian rings, this functor is left exact and a radical, see [14]. Its right derived functor H i a (−) is what is called the local cohomology functor with respect to a.…”