Exercise 3, § 3

Let $A$ be a Macaulay local ring. Recall (A, II, p. 186, exerc. 16) that an ideal of $A$ is said to be irreducible in $A$ if it is distinct from $A$ and is not the intersection of two ideals containing it strictly. Prove that the following conditions are equivalent:

(i) $A$ is a Gorenstein ring ; (ii) every ideal of $A$ generated by a maximal regular sequence is irreducible ; (iii) there exists a maximal regular sequence in $A$ generating an irreducible ideal. (Reduce to the case where $A$ is of dimension 0, and consider the socle $\mathrm{Hom}_A(\kappa_A, A)$ of $A$.)