Corollary 2

Let $A$ be a Macaulay ring and $n$ a positive integer; then $A[X_1, \ldots, X_n]$ and $A[[X_1, \ldots, X_n]]$ are Macaulay rings.

It is enough to treat the case $n = 1$. The ring $A[T]$ is noetherian (A, VIII, § 1, No. 4, cor. 1), and, for every field $k$, the ring $k[T]$ is a Macaulay ring (No. 5, example 2); consequently, the ring $A[T]$ is a Macaulay ring (prop. 10) and the ring $A[[T]]$ is a Macaulay ring (cor. 4 of prop. 9).