FI-gr-内射模
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摘要
本文引入了FI-gr-内射模及强FI-gr-内射模的概念,并说明它们与分次内射模之间的相互关系。证明了分次环R是分次QF环当且仅当每个分次模是强FI-gr-内射模;设R为左分次凝聚环,则l.FP-gr-dim(R)≤1当且仅当每个FI-gr-内射模是分次内射模。此外,还证明了l.gr-fiD(R)=sup{gr-pd(L)|L为FP-gr-内射模}。
In this paper,the notions of FI-gr-injective modules and strongly FI-gr-injective modules are introduced,and the relationship between them and graded injective modules is explained.It is proved that the graded ring Ris a gr-QF ring if and only if each graded module is a strongly FI-gr-injective module;suppose Ris a left gr-coherent ring,then l.FP-gr-dim(R)≤1 if and only if each FI-gr-injective module is a graded injective module.In addition,it is also proved that l.gr-fiD(R)=sup{gr-pd(L)|Lis an FP-gr-injective module}.
In this paper,the notions of FI-gr-injective modules and strongly FI-gr-injective modules are introduced,and the relationship between them and graded injective modules is explained.It is proved that the graded ring Ris a gr-QF ring if and only if each graded module is a strongly FI-gr-injective module;suppose Ris a left gr-coherent ring,then l.FP-gr-dim(R)≤1 if and only if each FI-gr-injective module is a graded injective module.In addition,it is also proved that l.gr-fiD(R)=sup{gr-pd(L)|Lis an FP-gr-injective module}.
引文
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[2] MADDOX B H.Absolutely pure modules[J].Proceedings of the American Mathematical Society,1967,18(1):155-158.DOI:10.2307/2035245.
[3] MAO L X,DING N Q.FI-injective and FI-flat modules[J].Journal of Algebra,2007,309(1):367-385.DOI:10.1016/j.jalgebra.2006.10.019.
[4] GAO Z H.On n-FI-Injective and n-FI-Flat Modules[J].Communications in Algebra,2012,40(8):2757-2770.DOI:10.1080/00927872.2011.585677.
[5] ASENSIO M J,RAMOS J A L,TORRECILLAS B.FP-gr-injective modules and gr-FC-rings[J].Algebra and Number Theory,1999:1-11.DOI:10.1201/9780203903889.ch1.
[6] YANG X Y,LIU Z K.FP-gr-injective modules[J].Mathematica Journal of Okayama University,2011,53:83-100.
[7] MAO L X.Strongly Gorenstein graded modules[J].Frontiers of Mathematics in China,2016,12(1):1-20.DOI:10.1007/s11464-016-0595-y.
[8] NSTSESCU C,OYSTAEYEN F V.Methods of Graded Rings[M].New York:Springer,2004:25-286.
[9] ROZAS J G,RAMOS J A L,TORRECILLAS B.On the existence of flat covers in R-gr[J].Communications in Algebra,2001,29(8):3341-3349.DOI:10.1081/AGB-100105025.
[10] OYONARTE L,TORRECILLAS B.Large subdirect products of graded modules[J].Communications in Algebra,1999,27(2):681-701.DOI:10.1080/00927879908826456.
[11] NSTSESCU C,OYSTAEYEN F V.Graded Ring Theory[M].New York:North-Holland Publishing Company,1982:52-61.
[12] MAO L X,DING N Q.Envelopes and covers by modules of finite injective and flat dimensions[J].Communications in Algebra,2007,35(3):833-849.DOI:10.1080/00927870601115757.
[13] HUSSIEN S E D S,EL-SEIDY E,TABARAK M E.Graded injective modules and graded quasi-frobenius rings[J].Applied Mathematical Sciences,2015,9(23):1113-1123.DOI:10.12988/ams.2015.4121036.
[14] KRASNOVA E N.Graded quasi-Frobenius rings[C]//Proceedings XII International Conference Algebra and Number Theory:Modern Problems and Application,dedicated to 80-th anniversary of Professor V.N.Latyshev.[S.l.:s.n.],2014:88-90.
[15]吴娜.FP-gr-投射维数和FP-gr-投射模[D].南京:南京师范大学,2013.
[16] NSTSESCU C.Strongly graded rings of finite groups[J].Communications in Algebra,1983,11(11):1033-1071.DOI:10.1080/00927872.1983.10487600.
[17]王芳贵.交换环与星型算子理论[M].北京:科学出版社,2006:104-173.
[18] ROTMAN J J.An introduction to homological algebra[M].New York:Springer,2009:418-421.
[19] KIRKMAN E,KUZMANOVICH J.Matrix subrings having finite global dimension[J].Journal of Algebra,1987,109(1):74-92.DOI:10.1016/0021-8693(87)90165-7.
[20] XU J Z.Flat covers of modules[M].New York:Springer,1996:27-28.