X-GORENSTEIN PROJECTIVE AND Y-GORENSTEIN INJECTIVE MODULES

Year 2011, Volume: 40 Issue: 4, 537 - 554, 01.04.2011

Fanyun Meng Qunxing Pan

Abstract

X-GORENSTEIN PROJECTIVE AND Y-GORENSTEIN INJECTIVE MODULES

Abstract

Let X be a class of right R-modules that contains all projective right R-modules. The notion of X-Gorenstein projective modules was introduced by D. Bennis and K. Ouarghi (X-Gorenstein projective modules, International Mathematical Forum 5 (10), 487–491, 2010). In this paper, we introduce Y-Gorenstein injective right R-modules and Y-Gorenstein flat left R-modules, where Y is a class of right R-modules that contains all injective right R-modules. We show that the principal results on Gorenstein modules remain true for X-Gorenstein projective right R-modules, Y-Gorenstein injective right R-modules and Y-Gorenstein flat left R-modules.

Keywords

X-Gorenstein projective modules, Y-Gorenstein injective modules, Y-Gorenstein flat modules, 2000 AMS Classification: 16 E 10

References

• Bennis, D. and Mahdou, N. Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2), –465, 2010.
• Bennis, D. and Ouarghi, K. X-Gorenstein projective modules, International Mathematical Forum 5 (10), 487–491, 2010.
• Christensen, L. W. Gorenstein Dimensions, Lecture Notes in Math. 1747 (Springer, Berlin. Heidelberg, 2000).
• Ding, N. Q., Li, Y. L. and Mao, L. X. Strongly Gorenstein flat modules, J. Aust. Math. Soc. , 323–338, 2009.
• Enochs, E. E. Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189– , 1981.
• Enochs, E. E. and Jenda, O. M. G. Gorenstein injective and Gorenstein projective modules, Math. Z. 220, 611–633, 1995.
• Enochs, E. E. and Jenda, O. M. G. Relative Homological Algebra, GEM 30 (Walter de Gruyter, Berlin-New York, 2000).
• Enochs, E. E., Jenda, O. M. G. and L´opez-Ramos, J. A. The existence of Gorenstein flat covers, Math. Scand. 94, 46–62, 2004.
• Enochs, E. E. and Oyonarte, L. Covers, envelopes and cotorsion theories (Nova Science Publishers, Inc, New York, 2002).
• Fieldhouse, D. J. Character modules, dimension and purity, Glasgow Math. J. 13, 144–146, G¨obel, R. and Trlifaj, J. Approximations and Endomorphism Algebras of Modules, GEM (Walter de Gruyter, Berlin-New York, 2006).
• Holm, H. Gorenstein homological dimensions, J. Pure Appl. Algebra 189, 167–193, 2004.
• Mao, L. X. and Ding, N. Q. Gorenstein F P -injective and Gorenstein flat modules, J. Algebra Appl. 7, 491–506, 2008.
• Megibben, C. Absolutely pure modules, Proc. Amer. Math. Soc. 26 (4), 561–566, 1970.
• Tamekkante, M. The right orthogonal class GP(R)⊥viaExt, arXiv: 0911.1272.