Covers and envelopes over Gorenstein rings

A module over a Gorenstein ring is said to be Gorenstein injective if it splits under all modules of finite projective dimension. We show that over a Gorenstein ring every module has a Gorenstein injective envelope. We apply this result to the group algebra ZpG (with G a finite group and Zp the ring of p-adic integers for some prime p) and show that ever finitely genecrated ZpG-module has a cover by a lattice. This gives a way of lifting finite dimensional representations of G over Z/(p) to modular representations of G over Zp