关于w-linked扩环

Let R ■ T be an extension of commutative rings.T is called w-linked over R if T as an R-module is a w-module.In the case of R ■ T ■ Q 0 （R）,T is called a w-linked overring of R.As a generalization of Wang-McCsland-Park-Chang Theorem,we show that if R is a reduced ring,then R is a w-Noetherian ring with w-dim（R） 1 if and only if each w-linked overring T of R is a w-Noetherian ring with w-dim（T ） 1.In particular,R is a w-Noetherian ring with w-dim（R） = 0 if and only if R is an Artinian ring.     

同态模Hom(A,B)的同调维数

本文从研究函子(?)与Hom的联系入手,来考虑求Hom(A,B)的弱维数与投射维数。当K为域时,且条件(a)[R:K]∞,A是有限生成右R模;(b)·[R:K]<∞,S是右凝聚代数:(c)[S:K]∞,R是右Noether代数,有一成立得到1.wd_R(?)SHom(A,B)r.id_RA+1.wd_sB。     

本原环的Grothendieck群

设R为本原环,对应的忠实既约模为T,且soc(R)≠R,设R=R/soc(R).在文中证明了以下结果: (1)K_o(R)→K_o(R)是满同态,且当soc(R)≠0时,N=Ker(K_o(R)→K_o(R))是由[T]∈K_o(R)生成的循环子群. (2)若soc(R)=0,则存在一个本原环R_1,soc(R_1)≠0,使得R是R_1的同态象,且K_o(R_1)≌K_o(R)⊕N,其中N=Ker(K_o(R_1)→K_o(R))是由[T]∈K_o(R_t)生成的循环子群.     

平坦的多项式剩余类环

本文证明了如果多项式的剩余类环 A=R[T]/fR[T]作为 R-模是平坦模,且R是约化环,则f是正规多项式.特别地,若R还是连通的,则f的首项系数是单位.也证明了弱整体有限的凝聚环是约化环,以及弱整体为有限的凝聚连通环是整环.     

一类弱整体维数为2的局部环上的Bass-Quillen问题Ⅱ

设R是U2环,即(R,m)是局部GCD整环,且存在u∈m-m2,使得R/(u)是赋值环,且Ru是Bézout整环.本文证明了若R是带有正规元素为u的U2环,且dim(R/(u))=1,则每个有限生成投射R[X1,...,Xn]模是自由模.由此得到了若R是广义伞环,则每个有限生成投射R[X1,...,Xn]模是自由模.     

Milnor方图中的w-凝聚性

整环R称为ω-凝聚整环,是指R的每个有限型理想是有限表现型的.本文证明了ω-凝聚整环是v-凝聚整环,且若(RDTF,M)是Milnor方图,则在Ⅰ型情形,R是ω-凝聚整环当且仅当D和T都是ω-整环,且T_M是赋值环;对于Ⅱ-型情形,R是ω-凝聚整环当且仅当D是域,[F:D]<∞,M是R的有限型理想,T是ω-凝聚整环,且R_M是凝聚整环.     

关于Mori整环的$w$-全局变换环的注记

Let R be a domain and let Rwg be the w-global transform of R. In this note it is shown that if R is a Mori domain, then the t-dimension formula t-dim（Rwg） = t-dim（R） - 1 holds.     

Cohn环与Grothendieck群

We introduce three classes of cohn rings C₁,C₂ and C₃). Let R and S are rings,φ : R→S is a ring homomorphism. We prove that R∈C_i if and only if R/J(R) ∈ C_i. In We also discuss the relation between the     

GCD整环与自反模

本文证明了凝聚整环是ＧＣＤ整环当且仅当秩为１的自反模是自由模．同时还得到有限弱整体维数的凝聚整环是ＧＣＤ整环当且仅当Ｐｉｃ（Ｒ）＝１．特别地，有限整体维数的Ｎｏｅｔｈｅｒ整环是ＵＦＤ当且仅当Ｐｉｃ（Ｒ）＝１．     

反向极限与模的FP-内射包

Let R be a ring with unit element, and A a left R-module. A is called FP-injective if Ext_R′(P, A)=0 for every finitely presented R-module P. Let E_R(A) denote the injective hull of A. In the paper we prove by means of inverse limit functor that any module A Over coherent ring R has the minimum FP-injective submodule e_R(A) in E_R(A) which contains A and can be defined as the FP-injective hull of A.