What is $Hom_A(A/I,M)$?

Question

Let $A$ be a commutative ring. Let $I$ be an ideal in $A$. Let $M$ be an $A$-module. It seems like $Hom_A(A/I,M)$ should measure $I$-torsion in $M$. Does $Hom_A(A/I,M)$ have a nice description or property?

Answer 1

$Hom_A(A/I,M)=\{\alpha \in Hom_A(A,M) : I \subseteq \ker(\alpha)\}=\{m \in M : mI=0\}$