Exercise 1, § 1

Let $A$ be a ring, $J$ an ideal of $A$, and $M$ an $A$-module. Prove that in order that $\mathrm{prof}_A(J; M) \geq 2$, it is necessary and sufficient that the mapping $M \to \mathrm{Hom}_A(J, M)$ deduced from the canonical injection of $J$ into $A$ be an isomorphism.