分次环上的分次w-模
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摘要
R=σ∈GRσ是有单位元1的交换的G-分次环(在G不需言明时就称R为分次环),并且引入了分次环上的分次w-模等相关概念.证明了:1)设J是R的有限生成分次理想,则J∈GVgr(R)当且仅当J∈GV(R);2)设M是分次模,σ∈G.若M是分次GV-无挠模(或分次GV-挠模),则M(σ)也是分次GV-无挠模(或分次GV-挠模);3)设M是分次模,且是w-模,N是M的分次子模,则N是分次w-模当且仅当N是w-模.特别地,R中的任何分次w-理想都是w-理想.
In this paper,R = σ∈GRσis a commutative G-graded ring with identity 1. We also call R a graded ring for short. Besides,graded w-modules and other related conceptions over a graded ring R are introduced. It is shown that: 1) let J be a finitely generated graded ideal of R. Then J is a graded GV-ideal if and only if J is a GV-ideal. 2) If M is a graded GV-torsion-free module( respectively,GV-torsion module),then the σ-suspended graded module M( σ) is also a graded GV-torsion-free module( respectively,GV-torsion module). 3) Let M be a graded w-module and N be a graded submodule of M. Then N is a graded w-module if and only if N is a w-module. Especially,a graded w-ideal of R is a w-ideal.
In this paper,R = σ∈GRσis a commutative G-graded ring with identity 1. We also call R a graded ring for short. Besides,graded w-modules and other related conceptions over a graded ring R are introduced. It is shown that: 1) let J be a finitely generated graded ideal of R. Then J is a graded GV-ideal if and only if J is a GV-ideal. 2) If M is a graded GV-torsion-free module( respectively,GV-torsion module),then the σ-suspended graded module M( σ) is also a graded GV-torsion-free module( respectively,GV-torsion module). 3) Let M be a graded w-module and N be a graded submodule of M. Then N is a graded w-module if and only if N is a w-module. Especially,a graded w-ideal of R is a w-ideal.
引文
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[2]GOTO S,YAMAGISHI K. Finite generation of noetherian graded rings[J]. Pro AMS,1983,89:41-43.
[3]LEE E K,PARK M H. On graded Krull overring of a graded Noetherian domain[J]. Bull Korean Math Soc,2012,49:205-211.
[4]HEINZER W. On Krull overrings of a Noetherian domain[J]. Pro AMS,1969,22:217-222.
[5]HUSSIEN S E D S,EL-SEIDY E,TABARAK M E. Graded injective modules and graded quasi-Frobenius rings[J]. Appl Math Sci,2015,23:1113-1123.
[6]NISHIMURA J. Note on integral closure of a noetherian integral domain[J]. Math Kyoto Univ,1976,16:117-122.
[7]PARK C H,PARK M H. Integral closure of a graded Noetherian domain[J]. J Korean Math,2011,48:449-464.
[8]HUANG Z Y. On the graded of modules over noetherian ring[J]. Commun Algebra,2008,36:3616-3631.
[9]ANDERSON D D,ANDERSON D F. Divisorial ideals and invertible ideals in a graded integral domains[J]. J Algebra,1982,76:549-569.
[10]ANDERSON D D,ANDERSON D F. Divisibility properties of graded domains[J]. Can J Math,1982,34:196-215.
[11]ANDERSON D F. Graded Krull domains[J]. Commun Algebra,1979,7:79-106.
[12]PARK M H. Integral closure of graded integral domains[J]. Commun Algebra,2007,35:3965-3978.
[13]WANG F G,MCCASLAND R L. On strong mori domains[J]. J Pure Appl Algebra,1999,135:155-165.
[14]YIN H Y,WANG F G,ZHU X S,et al. w-modules over commutative rings[J]. J Korean Math Soc,2011,48:207-222.
[15]NASTASESCU C,OYSTAEYEN F V. Methods of Graded Rings[M]. New York:Springer-Verlag,2004.
[16]NASTASESCU C,OYSTAEYEN F V. Graded Ring Theory[M]. New York:Springer-Verlag,1982.
[17]WNAG F G,KIM H. Foundations of Commutative Rings and Their Modules[M]. Singapore:Springer-Verlag,2016.
[18]OYONARTE L,TORRECILLAS B. Large subdirect products of graded modules[J]. Commun Algebra,1999,27:681-701.
[19]EKLOF P,TRLIFAJ J. How to make ext vanish[J]. Bull London Math Soc,2001,33:41-51.