×

Covers and envelopes over gr-Gorenstein rings. (English) Zbl 0942.16049

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.

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16D90 Module categories in associative algebras
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Asensio, M. J.; Lopez Ramos, J. A.; Torrecillas, B., Gorenstein gr-injective and gr-projective modules, Comm. Algebra, 26, 225-240 (1998) · Zbl 0895.16020
[3] Auslander, M.; Buchweitz, R., The homological theory of maximal Cohen-Macaulay approximations, Mem. Soc. Math. France, 38, 5-37 (1989) · Zbl 0697.13005
[4] Garcia Rozas, J. R.; Torrecillas, B., Preserving and reflecting covers by functors: Applications to graded modules, J. Pure Appl. Algebra, 112, 91-107 (1996) · Zbl 0858.16025
[5] Eisenbud, D., Commutative Algebra with a View toward Algebraic Geometry (1995), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0819.13001
[6] Enochs, E., Injective and flat covers, envelopes and resolvents, Israel J. Math., 39, 189-209 (1981) · Zbl 0464.16019
[7] Enochs, E.; Jenda, O., Gorenstein injective and projective modules, Math. Z., 220, 303-310 (1995) · Zbl 0845.16005
[9] Enochs, E.; Jenda, O.; Torrecillas, B., Gorenstein flat modules, J. Nanjing Univ. Math., 10, 1-9 (1993) · Zbl 0794.16001
[10] Enochs, E.; Jenda, O.; Xu, J., Covers and envelopes over Gorenstein rings, Tsukuba J. Math., 20, 487-503 (1996) · Zbl 0895.16001
[11] Enochs, E.; Xu, J., Gorenstein flat covers of modules over Gorenstein rings, J. Algebra, 181, 288-313 (1996) · Zbl 0847.16003
[13] Hermann, M.; Ikeda, S.; Orbanz, U., Equimultiplicity and Blowing Up (1988), Springer-Verlag: Springer-Verlag New York/Berlin
[14] Iwanaga, Y., On rings with finite self-injective dimension, II, Tsukuba J. Math., 4, 107-113 (1980) · Zbl 0459.16011
[15] Iwanaga, Y., On rings with finite self-injective dimension≤1, Osaka J. Math., 15, 33-46 (1978) · Zbl 0402.16017
[16] Jensen, C., On the vanishing of lim←, J. Algebra, 15, 151-166 (1970) · Zbl 0199.36202
[17] Nastasescu, C.; Van Oystaeyen, F., Graded Ring Theory. Graded Ring Theory, North-Holland Math. Library, 28 (1982), North-Holland: North-Holland Amsterdam · Zbl 0494.16001
[18] Nastasescu, C.; Van Den Bergh, M.; Van Oystaeyen, F., Separable functors applied to graded rings, J. Algebra, 123, 397-413 (1989) · Zbl 0673.16026
[19] Xu, J., Flat Covers of Modules. Flat Covers of Modules, Lecture Notes in Math., 1634 (1996), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0860.16002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.