I need to prove the following statment (actually a special case of it).
Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then $\operatorname{grade}(I,M)\geq 2$ if and only if the homomorphism $M\rightarrow$Hom$_R(I,M)$ given by $m\mapsto(i\mapsto im)$ is an isomorphism.
This is Exercise 1.2.24 in Bruns and Herzog, Cohen-Macaulay Rings.