Injective Envelopes and (Gorenstein) Flat Covers

Details Injective Envelopes and (Gorenstein) Flat Covers Injective Envelopes and (Gorenstein) Flat Covers

2009-09-13

arxiv.org/abs/0909.2415v2

Mathematics

Rings and Algebras

17 pages. It has been accepted for publication in Algebras and Representation Theory

Scientific paper

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.

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