Assassins and torsion functors

TL;DR

This paper investigates the relationship between associated primes of modules and their torsion submodules with respect to an ideal, providing conditions that link the associated primes of the module, its torsion part, and the quotient.

Contribution

It establishes new conditions characterizing the associated primes of modules and their torsion submodules in relation to an ideal, clarifying their interconnections.

Findings

Conditions for associated primes of torsion submodules

Conditions for associated primes of quotient modules

Characterizations involving weakly associated primes

Abstract

Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $\Gamma_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.