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Coherent ring
About: Coherent ring is a research topic. Over the lifetime, 161 publications have been published within this topic receiving 3564 citations.
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TL;DR: In this paper, the closely related Gorenstein projective, Goren stein injective and 2-at dimensions of modules are studied, and a generalization of these results is given to give homological descriptions of the GORNE dimensions over arbitrary associative rings.
762 citations
Book•
01 Jan 1996
TL;DR: Flat covers and cotorsion envelopes have been used in commutative rings as mentioned in this paper, where flat covers are used to cover the cotorion of envelopes.
Abstract: Envelopes and covers.- Fundamental theorems.- Flat covers and cotorsion envelopes.- Flat covers over commutative rings.- Applications in commutative rings.
497 citations
TL;DR: Using the dual of a categorical definition of an injective envelope, injective covers can be defined and shown to exist for all modules over a regular local ring of dimension 2 as discussed by the authors.
Abstract: Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension 2. Using injective covers, minimal injective resolvents can be defined.
472 citations
366 citations
TL;DR: In this paper, Ding projective, Ding injective and Ding flat modules are defined as analogs to Enochs' Gorenstein projective and Gorentein flat modules.
Abstract: An n-FC ring is a left and right coherent ring whose left and right self-FP-injective dimension is n. The work of Ding and Chen shows that these rings possess properties which generalize those of n-Gorenstein rings. In this paper we call a (left and right) coherent ring with finite (left and right) self-FP-injective dimension a Ding-Chen ring. In the case of Noetherian rings, these are exactly the Gorenstein rings. We look at classes of modules we call Ding projective, Ding injective and Ding flat which are meant as analogs to Enochs' Gorenstein projective, Gorenstein injective and Gorenstein flat modules. We develop basic properties of these modules. We then show that each of the standard model structures on Mod-R, when R is a Gorenstein ring, generalizes to the Ding-Chen case. We show that when R is a commutative Ding-Chen ring and G is a finite group, the group ring R[G] is a Ding-Chen ring.
116 citations