Covers and envelopes over gr-Gorenstein rings
A (possibly non-commutative) ring \(R\) is called Gorenstein if \(R\) is Noetherian and has finite injective dimensions on both sides. In this paper, the authors study a relative homological theory for graded modules over graded (briefly, gr-) Gorenstein rings, continuing their recent work on this topic. The emphasis in the paper is on the existence of envelopes and covers for the classes of graded modules. For a graded ring \(R\), the category \(R\)-gr of all graded left \(R\)-modules is a Grothendieck category, and the injective and projective objects in this category are called gr-injective and gr-projective, respectively. The authors introduce the concepts of Gorenstein gr-injective and gr-projective modules which are analogous to the ungraded definitions over Gorenstein rings. It is shown that Gorenstein gr-injective envelopes exist for gr-modules over gr-Gorenstein rings. The existence of Gorenstein gr-projective covers for finitely generated graded modules over certain gr-Gorenstein rings is also studied. Among other results, the authors consider the interesting relation between the existence of envelopes and covers in the category of graded modules and the category of modules over a graded Gorenstein ring.
scientific article; zbMATH DE number 1309122
scientific article