Injective Envelopes and (Gorenstein) Flat Covers

Edgar E. Enochs; Zhaoyong Huang

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arXiv · 0909.2415

Injective Envelopes and (Gorenstein) Flat Covers

Edgar E. Enochs·Zhaoyong Huang

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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.

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BibTeX RIS

Edgar E. Enochs, Zhaoyong Huang. 2011-03-20. Injective Envelopes and (Gorenstein) Flat Covers. https://arxiv.org/abs/0909.2415

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