G-集分次模的分次自同态环

设G是任意群,本文给出了G-集G／H-分次模的分次自同态环的刻画．特别地,对我们证得N（H）／H-分次自同态环END（G／H,R）－gr（M）等于分次环ENDR（M）N（H）／H.     

G-分次环与G-集的冲积(Smash Product)

对任意群G,[1]中研究了G-分次环与可迁有限G-集的冲积.在本文中我们对任意可迁G-集A,讨论了G-分次环R与G-集A的冲积,从而推广了[2][3]中给出的关于G-分次环与群G的冲积的主要结果.     

G-分次环与G-集的冲积(Smash Product)

对任意群G,[1]中研究了G-分次环与可迁有限G-集的冲积.在本文中我们对任意可迁G-集A,讨论了G-分次环R与G-集A的冲积,从而推广了[2][3]中给出的关于G-分次环与群G的冲积的主要结果.     

强分次环与非分次模范畴

设R是强群G-分次环,日是G的指数有限的子群.本文讨论强分次环R上一般非分次模的一些性质.首先证明一个与非分次R-模和R(H)-子模有关的Maschke-type定理,然后证明从模范畴R(H)-mod到R-mod的函子HomR(H)(R,-)与R(?)R(H)-是一对自然同构的函子以及等价条件.     

半群S-分次环与冲积R#S~*

设S为任意半群,本文以S-集为基础,讨论了S-集分次模的一些性质并得到范畴(S,A,R)-gr的一个有限生成投射生成子集合;对于冲积R#S*,主要证明了在一定条件下, unital左R#S*-模范畴和分次R-模范畴是同构的.     

有限群分次环上的余挠对研究

本文研究了余挠对在有限群分次和非分次情况下的联系.利用分次理论以及相对同调,我们首先研究了R是任意环G是有限群的情况下,余挠对在R-模范畴以及斜群环S=R*G-模范畴之间的关系;然后我们研究了R-gr范畴中刚性余挠对的等价刻画,同时给出了余挠对在R-gr范畴与R-模范畴之间的关系,其中R是G分次环,群G是有限群且|G|^-1∈R.     

G-分次环中一个定理的推广

本推广单分次摸情形下的G-分次环的一个定理．     

分次环与模范畴上的伴随函子

设 G是有限群 ,R是强 G-分次环 .本文证明了 R Re-与 Hom Re(R,- )都是从模范畴 R - mod到 Re- mod的“纯量”限制函子 F的伴随函子 ,并且两个函子 R Re-和Hom Re(R,- )是自然同构的 .     

Morita对偶和Smash积

设G是有限群,e为G的单位元,R=是有单位元的G-型分次环,T=R_e,R_U是极小内射余生成子.本文中,我们证明了R有左Morita对偶当且仅当Smash积R#G有左Morita对偶.设H是G的(正规)子群,若R有左Morita对偶,则R~((H))#H(R_((G/H))#(G/H))有左Morita对偶。当R是强分次环时,T有左Morita对偶当且仅当R有左Morita对偶当且仅当R#G有左Morita对偶.     

分次Armendariz环与P.P.环

引进分次Armendariz环的概念,讨论了分次环R= n∈2Rn及由它导出的非分次环R,R0,及R[x]之间关于Armendariz环性质的关系,并推广了[8]的结论,得到在R= n∈ZRn是Z-型正分次环的前提下,若R是分次Armendariz,分次正规环,则R是P．P环（Bear环）当且仅当R是分次P．P．环（分环Baer环）。     

局部单位元半群分次环

设S为有限局部单位元半群,R为S—分次环.首先定义了S—分次环R在半群S上的冲积R#S*,证明了模范畴R#S*-M od与分次模范畴(S,R)-g r之间的等价性,并进一步研究了局部单位元半群分次环的分次Jacobson根及其相关的自反根的关系,得到重要关系式J(R#S*)=JS(R)#S*及Jref(R)=(J(R#S*))↓=JS(R).     

同调方程

设Ｒ是左、右凝聚环,Ｒ~ωＲ是一个忠实平衡自正交双模．对有限表现左Ｒ－模Ａ和正整数ｎ,本文研究了形如,　的同调方程．给出了模范畴为有限表现右Ｒ-模｝是子模闭的充要条件,并举例说明了该模范畴并非总是子模闭的,     

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ?(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.     

One-dimensional local rings with reduced associated graded ring and their Hilbert functions

Let A be a one-dimensional reduced local ring with finite normalization. Let G(A) be the associated graded ring of A. In this paper we analyse the two conditions: Proj (G(A)) reduced and G(A) reduced together with their relations with the equality H(n)=H_R (n), where H(n) and H_R (n) are respectively the Hilbert function of the ring A and of the local ring R of G(A)_red=G(A)/nil (G(A)) at its homogenous maximal ideal. As a consequence of our results we get a class of ordinary singularities with H(n) locally decreasing for any embedding dimension H(1) greater then 4.     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.     

Goldie dimension of a sum of modules

The formula dim(A+B)=dim(A)+dim(B)-dim(A∩B) works when ‘dim’ stands for the dimension of subspaces A,B of any vector space. In general, however, it does no longer hold if 'dim' means the uniform (or Goldie) dimension of submodules A,B of a module M over a ring R, and in fact the left hand side may be infinite while the right hand side is finite. In this paper we shall give a characterization of those modules M in which the formula holds for any two submodules A,B, as well as some conditions in the ring R which guarantee that dim(A+B) is finite whenever A and B are finite dimensional R-modules.     

正则WB-环

引进了WB-环,研究了正则环为WB-环的等价刻画．如果A是正则环R上的有限生成投射右模而且M_n(R)都是WB-环(n∈N),若B,C是任何右R-模而且A⊕B≌A⊕C,证明了存在正交理想I,J,使得B/BI■~⊕C/CI且C/CJ■~⊕B/BJ．这也给出了QB-环上新的模比较性质．     

H-SEPARABLE RINGS AND THEIR HOPF-GALOIS EXTENSIONS

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＡｂｅａｒｉｎｇａｎｄＢａｓｕｂｒｉｎｇｏｆＡ．ＡｉｓｃａｌｅｄａｎｅｘｔｅｎｓｉｕｏｎｏｆＢｄｅｎｏｔｅｄｂｙＡ／ＢｉｆＢａｎｄＡａｄｍｉｔｔｈｅｓａｍｅｉｄｅｎｔｉｔｙ．ＦｏｒａｎｅｘｔｅｎｓｉｏｎＡ／Ｂ，ｗｅ...