Example 2

Let $A$ be a Gorenstein ring and let $J$ be an ideal of $A$, generated by an $A$-regular sequence $x$. The quotient ring $A/J$ is a Gorenstein ring: in fact, for every maximal ideal $m$ of $A$ containing $J$, the image in $A_m$ of the sequence $x$ is $A_m$-regular and generates the ideal $J_m$, so that $A_m/J_m$ is a Gorenstein ring by Cor. of Prop. 7 (No. 4).