Corollary 3

Let $A$ be a noetherian ring and $J$ a completely secant ideal of $A$. If $A$ is a Macaulay ring (resp. a Gorenstein ring), the same is true of $A/J$.

Let in fact $m$ be a maximal ideal of $A$ containing $J$. The ideal $J_m$ of $A_m$ is generated by an $A_m$-regular sequence; hence $(A/J)_m$ is a Macaulay ring (resp. a Gorenstein ring) by Example 6 of § 2, No. 5 (resp. Example 2 of § 3, No. 9).