Embedding of modules over Gorenstein rings

Abstract:

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring. The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated A-module can be embedded in a finitely generated A-module T which is pointwise of finite injective dimension (i.e. all localizations of T are of finite injective dimension). The second result is that if A is Cohen-Macaulay, then every finitely generated A-module can be embedded in a finitely generated A-module of finite projective dimension if and only if A is Gorenstein.