# 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
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## 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.