交换环的w-弱finitistic维数的注记
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摘要
设R是交换环,M是R-模.引入了模M的w-投射维数w-pd_R(M)和环R的w-弱finitistic维数w-f PD(R).给出w-f PD(R)=0的充分必要条件.证明了若R是w-凝聚环,M是有限表现R-模,则M有w-投射分解…→P_n→P_(n-1)→…→P_1→P_0→M→0,其中P_i是有限型的w-投射模,这里i=0,1,….最后,证明了若R是w-半遗传环,w-f PD(R)#1.
Let R be a commutative ring and M be an R-module.This paper introduces and studies the w-projective dimension w-pd_R(M)of an R-module M and the w-weak finitistic dimension w-f PD(R)of R.The sufficient and necessary condition of w-f PD(R)=0 is given.As an application,it is shown that if R is a w-coherent ring,M is of finitely presented type,then M has a w-projective resolution…→P_n→P_(n-1)→…→P_1→P_0→M→0,where P_iis w-projective of finite type for i=0,1,….Finally,if R is w-semi-hereditary,then w-f PD(R)#1.
Let R be a commutative ring and M be an R-module.This paper introduces and studies the w-projective dimension w-pd_R(M)of an R-module M and the w-weak finitistic dimension w-f PD(R)of R.The sufficient and necessary condition of w-f PD(R)=0 is given.As an application,it is shown that if R is a w-coherent ring,M is of finitely presented type,then M has a w-projective resolution…→P_n→P_(n-1)→…→P_1→P_0→M→0,where P_iis w-projective of finite type for i=0,1,….Finally,if R is w-semi-hereditary,then w-f PD(R)#1.
引文
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[2]WANG F G,MCCASLAND R L.On Strong Mori Domains[J].Journal of Pure and Applied Algebra,1999,135(2):155-165.
[3]WANG F G,KIM H.Two generalizations of projective modules and their applications[J].Journal of Pure and Applied Algebra,2015,219(6):2099-2123.
[4]WANG F G,QIAO L.The w-weak global dimension of commutative ring[J].Bulletin of the Korean Mathematics Society,2015,52(4):1327-1338.
[5]FUSACCHIA G.Injective modules and semistar operations[J].Journal of Pure and Applied Algebra,2012,216:77-90.
[6]KIM H,KIM E S,PARK Y S.Injective modules over strong Mori domains[J].Houston Journal of Mathematics,2008,34:349-360.
[7]王芳贵,张俊.w-Noether环上的内射模[J].数学学报,2010,53(6):1119-1130.
[8]YIN H,WANG F G,ZHU X S,et al.w-modules over commutative rings[J].Journal of the Korean Mathematical Society,2011,48(1):207-222.
[9]WANG F G.On w-projective modules and w-flat modules[J].Algebra Colloquium,1997,4(1):111-120.
[10]WANG F G.有限表现型模和w-凝聚环[J].四川大学学报(自然科学版),2010:33(1):1-9.
[11]赵国,汪明义.关于极大内射性的注记[J].四川大学学报(自然科学版),2005,21(1):1451-1458.
[12]王芳贵,汪明义,杨立英.交换环上的极大性内射模[J].四川师范大学学报(自然科学版),2010,33(3):277-285.
[13]王芳贵.交换环与星型算子理论[M].北京:科学出版社,2006.
[14]KIM H,WANG F G.On LCM-stable modules[J].Journal of Algebra and its Applications,2014,13(4):1350133.
[15]WANG F G,KIM H.w-injective modules and w-semi-hereditary rings[J].Journal of the Korean Mathematical Society,2014,51(3):509-525.