LAUSR

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