I am new to this topic and would appreciate little explanation.
Def: A commutative, unital ring $A$ is a cubic ring if $A$ is a free $\mathbb{Z}$-module of rank $3$.
Def : A cubic ring $A$ is called Gorenstein if the $A$-module $Hom(A, \mathbb{Z})$ is projective.
I am a bit confused here. Every module over a PID is projective. Also for any $A$-module $M$ we have $Hom_{A}(A, M)\cong M$, so in particular $Hom_{\mathbb{Z}}(\mathbb{Z}, A)\cong A$ but $A$ is a free module hence projective, but then $Hom_{\mathbb{Z}}(\mathbb{Z}, A)$ must be projective as well, or am I mixing up something?