JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (8): 44-48.doi: 10.6040/j.issn.1671-9352.0.2016.216

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On relative homological dimensions

XU Ai-min   

  1. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
  • Received:2016-05-15 Online:2016-08-20 Published:2016-08-08

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Abstract: It is shown that pdC(X)=pdC(Y) and fdD(X)=fdD(Y), where X and C are two classes of left R-modules, D a class of right R-modules and Y={M|M is X-filtered}. As an application, the(weak)Gorenstein global dimensions of special rings are computed.

Key words: relative projective dimension, (weak)Gorenstein global dimension, relative flat dimension

CLC Number: 

  • O153.3
[1] AUSLANDER M,BUCHWEITZ R O. The homological theory of maximal Cohen-Macaulay approximations[J]. Societe Mathematique de France Memoire, 1989, 38:5-37.
[2] EKLOF P, TRLIFAJ J. How to make Ext vanish[J]. Bull London Math Soc, 2001, 33(1):41-51.
[3] ENOCHS E E, JENDA O M G. Relative homological algebra[M] // GEM 30. New York: Walter de Gruyter, 2000.
[4] GObel R, TRLIFAJ J. Approximations and endomorphism algebras of modules [M] // GEM 41. Berlin-New York: Walter de Gruyter, 2006.
[5] LIDIA A H, OCTAVIO M. Homological dimensions in cotorsion pairs[J]. Illinois J Math, 2009, 53(1):251-263.
[6] YANG Gang, LIU Zhongkui. Gorenstein flat covers over GF-closed rings[J]. Comm Algebra, 2012, 40(5):1632-1640.
[7] STENSROM B. Rings of Quotients[M]. Berlin: Springer-Verlag, 1975.
[8] BENNIS D, MAHDOU N. Global Gorenstein dimensions[J]. Proc Amer Math Soc, 2010, 138(2):461-465.
[9] ROTMAN J J. An introduction to homological algebra[M]. New York: Academic Press, 1979.
[10] XU J Z, CHENG F C. Homological dimensions over non-commutative semi-local rings[J]. J Algebra, 1994, 169(3):679-685.
[11] DING Nanqing, CHEN Jianlong. The homological dimensions of simple modules[J]. Bull Aust Math Soc, 1993, 48(48):265-274.
[12] LAM T Y. Lectures on Modules and Rings[M]. NewYork: Springer-Verlag, 1999.
[13] MAO Lixin, DING Nanqing. FP-projective dimensions[J]. Comm.Algebra, 2005, 33(4):1153-1170.
[14] BENNIS D. Rings over which the class of Gorenstein flat modules is closed under extensions[J]. Comm Algebra, 2009, 37(3):855-868.
[15] HOLM H. Gorenstein homological dimensions[J]. J Pure Appl Algebra, 2004, 189(1-3):167-193.
[16] KRAUSE H, RINGEL C M. Infinite length modules[M]. Berlin: Birkhäuser, 2000.
[17] ENOCHS E E, IACOB A, JENDA O M G. Closure under transfinite extensions[J]. Illinois J Math, 2007, 51(2):561-569.
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