Exercise 3, § 9

Let $A$ be a ring admitting a dualizing module $\Omega$.

a) In order that $\Omega$ be isomorphic to an ideal $I$ of $A$, it is necessary and sufficient that $A_p$ be a Gorenstein ring for every minimal prime ideal $p$ of $A$ (observe that $\Omega$ is isomorphic to a submodule of $\bigoplus_p \Omega_p$).

b) When these conditions are satisfied, prove that $I$ is equal to $A$ or of height 1, and that $A/I$ is a Gorenstein ring (calculate the depth of $A/I$, then the $A/I$-module $\mathrm{Ext}_A^1(A/I, I)$).

c) Prove that a factorial ring admitting a dualizing module is a Gorenstein ring.