Injective Envelopes and (Gorenstein) Flat Covers

TL;DR

This paper characterizes left Noetherian rings through properties of injective envelopes and flat covers, establishing equivalences involving Gorenstein flat modules and their envelopes, with implications for ring theory and module duality.

Contribution

It provides new characterizations of Noetherian rings based on the duality properties of injective envelopes and flat covers, especially in the Gorenstein context.

Findings

Injective envelope of $_RR$ is Gorenstein flat iff all Gorenstein flat modules have Gorenstein flat envelopes.

The flat dimension of injective envelopes of Gorenstein flat modules is bounded by that of $_RR$.

Multiple equivalent conditions involving Gorenstein flatness and injective envelopes characterize Noetherian rings.

Abstract

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left $R$-module is at most the flat dimension of the injective envelope of $_RR$. Then we get that the injective envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the injective envelope of every flat left $R$-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left $R$-module is injective, and if and only if the opposite version of one of these conditions is satisfied.