Example

Corollary 3 applies in particular in the following two situations:

a) Consider a noetherian integrally closed ring $A$, a separable extension $L$ of its field of fractions, of finite degree $n$ such that $n1_A$ is invertible in $A$, and take for $B$ the integral closure of $A$ in $L$ (V, $§ 1$, No. 6, Corollary 1 of Proposition 18).

b) Consider a noetherian integrally closed ring $B$ and a finite group $G$ of automorphisms of $B$, such that $\mathrm{Card}(G) 1_B$ is invertible in $B$. Take for $A$ the ring of elements of $B$ invariant under the action of $G$. Let us verify that we are in a particular case of a). The group $G$ operates on the field of the

In particular, if B is a Macaulay ring, the same is true of A.