Gornstein injective covers and envelopes over rings that satisfy the Auslander condition
Main Article Content
Abstract
It was recently proved ([12]) that the class of Gorensteininjective left R-modules is both covering and enveloping over a two sided noetherian ring R with the property that the character modules of the Gorenstein injective left R-modules are Gorenstein flat. It was also proved that over the same type of rings, the class of Gorenstein at right R-modules is preenveloping ([11]). We prove here that if R is a two sided noetherian ring R such that R satises the Auslander condition and has nite nitistic left injective dimension then R has the desired property: the character module of any Gorenstein injective is Gorenstein flat.
Article Details
How to Cite
Iacob, A.
(2016).
Gornstein injective covers and envelopes over rings that satisfy the Auslander condition.
Acta Mathematica Universitatis Comenianae, 85(1), 165-172.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/228/297
Issue
Section
Articles
References
[1] D. Bennis and N. Mahdou. Strongly Gorenstein projective, injective, and flat modules. J. Pure Apl. Algebra 210:437-445, 2007
[2] L. Christensen and A. Frankild and H. Holm. On Gorenstein projective, injective, and flat modules. A functorial description with applications. J. Algebra 302(1):231-279, 2006.
[3] E. Enochs and Z. Huang Injective envelopes and (Gorenstein) flat covers. Alg. and Represent. Theory 15:1131-1145, 2012.
[4] E. Enochs and A. Iacob. Gorenstein injective covers and envelopes over noetherian rings. Proc. of the American Math. Soc. 143(1): 5-12, 2015.
[5] E. Enochs and O.M.G. Jenda. Relative Homological Algebra. De Gruyter Exposition in Math.
[6] S. Estrada and A. Iacob and S. Odabasi. Gorenstein projective and flat (pre)covers. submitted
[7] H. Holm and P. Jorgensen. Cotorsion pairs induced by duality pairs. J. Commut. AlgebraI (4): 621-633, 2009
[8] H. Holm. Gorenstein homoogical dimensions. J. Pure Appl. Algebra, 189(1): 167193, 2004.
[9] Z. Huang. On Auslander type conditions of modules. preprint.
[10] H. Huang and Z. Huang. The Auslander type conditions of triangular matrix rings. Sci. China Math 55(8): 16471654, 2012.
[11] A. Iacob. Gorenstein flat preenvelopes Osaka J. Math., to appear.
[12] A. Iacob. Gorenstein injective envelopes and covers over two sided noetherian rings. submitted
[13] P. orgensen. Existence of Gorenstein projective resolutions and Tate cohomology. J. Eur. Math. Soc. 9:59{76, 2007.
[14] J.R. Garca Rozas. Covers and evelopes in the category of complexes of modules. CRC Press LLC 1999.
[15] H. Yan. Strongly cotorsion (torsion-free) modules and cotorsion pairs. Bull. Korean Math. Soc. 5: 10411052, 2010.
[2] L. Christensen and A. Frankild and H. Holm. On Gorenstein projective, injective, and flat modules. A functorial description with applications. J. Algebra 302(1):231-279, 2006.
[3] E. Enochs and Z. Huang Injective envelopes and (Gorenstein) flat covers. Alg. and Represent. Theory 15:1131-1145, 2012.
[4] E. Enochs and A. Iacob. Gorenstein injective covers and envelopes over noetherian rings. Proc. of the American Math. Soc. 143(1): 5-12, 2015.
[5] E. Enochs and O.M.G. Jenda. Relative Homological Algebra. De Gruyter Exposition in Math.
[6] S. Estrada and A. Iacob and S. Odabasi. Gorenstein projective and flat (pre)covers. submitted
[7] H. Holm and P. Jorgensen. Cotorsion pairs induced by duality pairs. J. Commut. AlgebraI (4): 621-633, 2009
[8] H. Holm. Gorenstein homoogical dimensions. J. Pure Appl. Algebra, 189(1): 167193, 2004.
[9] Z. Huang. On Auslander type conditions of modules. preprint.
[10] H. Huang and Z. Huang. The Auslander type conditions of triangular matrix rings. Sci. China Math 55(8): 16471654, 2012.
[11] A. Iacob. Gorenstein flat preenvelopes Osaka J. Math., to appear.
[12] A. Iacob. Gorenstein injective envelopes and covers over two sided noetherian rings. submitted
[13] P. orgensen. Existence of Gorenstein projective resolutions and Tate cohomology. J. Eur. Math. Soc. 9:59{76, 2007.
[14] J.R. Garca Rozas. Covers and evelopes in the category of complexes of modules. CRC Press LLC 1999.
[15] H. Yan. Strongly cotorsion (torsion-free) modules and cotorsion pairs. Bull. Korean Math. Soc. 5: 10411052, 2010.