Exercise 8, § 9

Let $A$ be a local Gorenstein ring, $a$ a nonzero ideal of height zero, and $b$ the annihilator of $a$.

a) Prove that the ideal $b$ is of height zero, and that the $A$-module $A/b$ has no immersed associated prime ideals (observe that every associated prime ideal of $A/b$ is associated to $A$).

b) Suppose that the $A$-module $A/a$ has no immersed associated prime ideals; prove that $a$ is the annihilator of $b$ (reduce to the case where $\dim(A) = 0$, and use Matlis duality).

c) For $A/a$ to be a Macaulay ring, it is necessary and sufficient that the same be true of $A/b$ (use the cor. of th. 1 and that of prop. 3). If these conditions are satisfied, the $A/a$-module $b$ is dualizing.