Example 1

For every multiplicative subset $S$ of a Gorenstein ring $A$, the ring of fractions $S^{-1}A$ is a Gorenstein ring: in fact, let $q$ be a maximal ideal of $S^{-1}A$; it is of the form $S^{-1}p$, where $p$ is a prime ideal of $A$ not meeting $S$. Let $m$ be a maximal ideal of $A$ containing $p$; then the ring $B = (S^{-1}A)_q$ is isomorphic to $A_p$ (II, § 2, No. 5, Prop. 11), hence to a ring of fractions of $A_m$, and consequently satisfies $\mathrm{di}_B(B) < +\infty$ (No. 2, Cor. 1 of Prop. 3).