正合零因子下模的G_C-同调维数
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摘要
设R是具有单位元的交换Noether环,C是半对偶化模,x是R上的正合零因子.考虑正合零因子下模的G_C-同调维数,证明了若M是G_C-投射(内射,平坦)R-模,则M/(xM)是G_C/(xC)-投射(内射,平坦)R/(xR)-模.对DC-投射(内射)R-模可得类似结论.
Let R be a commutative ring with identity,C be a semidualizing R-module and x be an exact zero-divisor over R.We considered the G_C-homological dimensions of modules under exact zero-divisors,and proved that if M was G_C-projective(injective,flat)R-module,then M/(xM)was G_C/(xC)-projective(injective,flat)R/(xR)-module.And the similar conclusions could be obtained for DC-projective(injective)R-modules.
Let R be a commutative ring with identity,C be a semidualizing R-module and x be an exact zero-divisor over R.We considered the G_C-homological dimensions of modules under exact zero-divisors,and proved that if M was G_C-projective(injective,flat)R-module,then M/(xM)was G_C/(xC)-projective(injective,flat)R/(xR)-module.And the similar conclusions could be obtained for DC-projective(injective)R-modules.
引文
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[8] Holm H,Jrgensen P.Semi-dualizing Modules and Related Gorenstein Homological Dimensions[J].Journal of Pure and Applied Algebra,2006,205(2):423-445.
[9] Sather-Wagstaff S,Sharif T,White D.Tate Cohomology with Respect to Semidualizing Modules[J].Journal of Algebra,2009,324(9):2336-2368.
[10] Takahashi R,White D.Homological Aspects of Semidualizing Modules[J].Mathematica Scandinavica,2010,106(1):5-22.
[11] Henriques I B D A,ega L M.Free Resolutions over Short Gorenstein Local Rings[J].Mathematische Zeitschrift,2011,267(3/4):645-663.
[12] Bergh P A,Celikbas O,Jorgensen D A.Homological Algebra Modulo Exact Zero-Divisors[J].Kyoto Journal of Mathematics,2014,54(4):879-895.
[13] Amanzadeh E,Dibaei M T.Auslander Class,GC,and C-Projective Modules Modulo Exact Zero-Divisors[J].Communications in Algebra,2015,43(10):4320-4333.
[14] Dibaei M T,Gheibi M.Sequence of Exact Zero-Divisors[J/OL].2012-04-23.http://cn.arxiv.org/abs/1112.2353.
[15] Holm H,White D.Foxby Equivalence over Associative Rings[J].Journal of Mathematics of Kyoto University,2007,47(4):781-808.
[16] DING Nanqing,LI Yulin, MAO Lixin.Strongly Gorenstein Flat Modules[J].Journal of the Australian Mathematical Society,2009,86(3):323-338.
[17] MAO Lixin,DING Nanqing.Gorenstein FP-Injective and Gorenstein Flat Modules[J].Journal of Algebra and Its Applications,2008,7(4):491-506.
[18] Gillespie J.Model Structures on Modules over Ding-Chen Rings[J].Homology,Homotopy and Applications,2010,12(1):61-73.
[19] ZHANG Chunxia,WANG Limin,LIU Zhongkui.Ding Projective Modules with Respect to a Semidualizing Module[J].Bulletin of the Korean Mathematical Society,2015,51(2):339-356.
[20] Christensen L W.Semi-dualizing Complexes and Their Auslander Categories[J].Transactions of the American Mathematical Society,2001,353(5):1839-1883.
[2] Enochs E E,Jenda O M G.Gorenstein Injective and Projective Modules[J].Mathematische Zeitschrift,1995,220(1):611-633.
[3] Enochs E E,Jenda O M G,Torrecillas B.Gorenstein Flat Modules[J].Journal of Nanjing University:Mathematical Biquarterly,1993,10(1):1-9.
[4] Enochs E E,Jenda O M G.Relative Homological Algebra[M].Berlin:De Gruyter,2000.
[5] Christensen L W.Gorenstein Dimensions[M].Berlin:Springer,2000.
[6] Holm H.Gorenstein Homological Dimensions[J].Journal of Pure and Applied Algebra,2004,189(1/2/3):167-193.
[7] Foxby H B.Gorenstein Modules and Related Modules[J].Mathematica Scandinavica,1972,31:267-284.
[8] Holm H,Jrgensen P.Semi-dualizing Modules and Related Gorenstein Homological Dimensions[J].Journal of Pure and Applied Algebra,2006,205(2):423-445.
[9] Sather-Wagstaff S,Sharif T,White D.Tate Cohomology with Respect to Semidualizing Modules[J].Journal of Algebra,2009,324(9):2336-2368.
[10] Takahashi R,White D.Homological Aspects of Semidualizing Modules[J].Mathematica Scandinavica,2010,106(1):5-22.
[11] Henriques I B D A,ega L M.Free Resolutions over Short Gorenstein Local Rings[J].Mathematische Zeitschrift,2011,267(3/4):645-663.
[12] Bergh P A,Celikbas O,Jorgensen D A.Homological Algebra Modulo Exact Zero-Divisors[J].Kyoto Journal of Mathematics,2014,54(4):879-895.
[13] Amanzadeh E,Dibaei M T.Auslander Class,GC,and C-Projective Modules Modulo Exact Zero-Divisors[J].Communications in Algebra,2015,43(10):4320-4333.
[14] Dibaei M T,Gheibi M.Sequence of Exact Zero-Divisors[J/OL].2012-04-23.http://cn.arxiv.org/abs/1112.2353.
[15] Holm H,White D.Foxby Equivalence over Associative Rings[J].Journal of Mathematics of Kyoto University,2007,47(4):781-808.
[16] DING Nanqing,LI Yulin, MAO Lixin.Strongly Gorenstein Flat Modules[J].Journal of the Australian Mathematical Society,2009,86(3):323-338.
[17] MAO Lixin,DING Nanqing.Gorenstein FP-Injective and Gorenstein Flat Modules[J].Journal of Algebra and Its Applications,2008,7(4):491-506.
[18] Gillespie J.Model Structures on Modules over Ding-Chen Rings[J].Homology,Homotopy and Applications,2010,12(1):61-73.
[19] ZHANG Chunxia,WANG Limin,LIU Zhongkui.Ding Projective Modules with Respect to a Semidualizing Module[J].Bulletin of the Korean Mathematical Society,2015,51(2):339-356.
[20] Christensen L W.Semi-dualizing Complexes and Their Auslander Categories[J].Transactions of the American Mathematical Society,2001,353(5):1839-1883.