Duality Between Injective Envelopes and Flat Covers over Noether Algebras

Abstract

It follows by Puuska that, over a commutative Noetherian ring $R$, a morphism $i:M\to I$ of $R$-modules is an injective envelope if and only if its Matlis dual $Ho{m}_R(i,E)$ is a flat cover for some injective cogenerator $E$, and equivalently for every injective $R$-module $E$. In this paper, we will generalize this result to non-commutative Noetherian algebras.