Exercise 10, § 2

(a) Let $\rho : A \to B$ be an injective homomorphism of noetherian rings, such that the A-submodule $A1_B$ of $B$ is a direct factor. Prove that if $B$ is a Macaulay ring, the same is true of $A$.

(Reduce to the case where $A$ is local and, by reasoning by induction on $\operatorname{prof}_A(B)$, where $\operatorname{Ass}(B)$ contains a maximal ideal. Now prove by considering a primary decomposition of 0 in $B$ that $B$ is then isomorphic to a product, and conclude by reasoning by induction on the number of maximal ideals of $B$.)

(b) Let $B$ be a Macaulay ring, $G$ a finite group of automorphisms of $B$ whose order is invertible in $B$. Prove that the ring of invariants of $G$ is a Macaulay ring.