群分次环的本原性及分次本原性

设A是一个用有限群G分次的环,本文给出了Smash积A＃G~*为本原环的一个判别准则,并证明了群分次环的每一个本原理想必定包含一个分次本原理想。作为一个推论,得到已被Cohen和Montgomery证实的Bergman猜想的另一个证明。此外还得到了A_1为本原环或单环的判别。     

Graded Reversibility in Integral Group Rings

We say that an algebra R, graded by a group, is graded reversible if ab=0 implies ba=0 where a,b are homogeneous elements of R. In this note, we study graded reversibility when R=ℤG (viewed as a ℤ-algebra) and the grading group is cyclic of order two. A complete characterization of when this occurs is obtained for some classes of groups (including dihedral groups) assuming a positive answer to an open question concerning group gradings.     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

分次环的分次Jacobson根

本文通过引入弱拟正则元的概念，对一般Ｍｏｎｏｉｄ分次环Ａ（未必有１）给出以内部元素刻划的分次Ｊａｃｏｂｓｏｎ根ＪＧ（Ａ）．证明当Ａ有１时，ＪＧ（Ａ）与通常定义的Ｊｇ（Ａ）相等．对ＪＧ（Ａ）性质的讨论，推广了最近的许多结果．作为应用，我们给出了Ａｒｔｉｎ分次环的全部基本结构定理．     

Orthogonal Graded Completion of Graded Semiprime Rings

For an associative gr-semiprime ring R with identity graded by a group, the orthogonal graded completion O ^gr(R) is constructed. A criterion for the orthogonal completeness of the maximal right graded quotient ring Q ^gr(R) is proved. The ring Q ^gr(R) need not be orthogonally complete, as opposed to the ungraded case.     

分次环的分次根

给出了建立分次环根的一般方法．作为其应用，建立了分次环的分次Brown—McCoy根，并给出了Brown—McCoy半单分次环的结构定理．     

Gabriel Dimension for Graded Rings

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕ _σ∈G R _σ with finite support if R _e has Gabriel dimension then has Gabriel dimension. Moreover, adding some lattice results, we prove that if has Gabriel dimension then also has Gabriel dimension. This research has been supported by grant MTM2005-3227 from MEC.     

Density Theorems for Graded Rings

The main purpose of this paper is to prove three density theorems for rings graded by semigroups and modules graded by acts over these semigroups with some cancellation conditions. In addition, the density theorem for superrings and supermodules is proved.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 27–49, 2003.     

Graded Endomorphism Rings and Equivalences

Abstract

Let R be a G-graded ring,M a G-graded Σ-quasiprojective R- module,and E = END _R (M) its graded ring of endomorphisms. For any subgroup H of G,we prove that certain full subcategories of G/H-graded R-modules associated with M are equivalent to a quotient category of G/H-graded E-modules determined by the idempotent G-graded ideal of E consisting of endomorphisms which factor through a finitely generated submodule of M. Properties and applications of these equivalences are also examined.     

具有非零分次基座的分次本原环的刻划

In this paper, we characterize the structure of group graded primitive rings having minimal graded one-sided ideals in terms of graded dual modules. We also show that two structure representations of this class of graded primitive rings are both in a sense uniquely determined by the graded primitive rings considered.     

On Minimal Graded One-sided Ideals and Graded Primitive Rings

In this paper, sturcture theorems are given for graded primitive rings with minimal graded one-sided ideals and arbitrary minimal graded one-sided ideals of general group-graded rings respectively.     

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）。     

Depth of Higher Associated Graded Rings

The depth of the associated graded ring of the powers of anideal I of a local ring R is studied. It is proved that thedepth of the associated graded ring of Iⁿ is asymptoticallyconstant when n tends to infinity, and this value is characterizedin terms of Valabrega–Valla conditions of I^m for somelarge integer m 0. As a corollary, a generalization is obtainedof the 2-dimensional algebraic version of the Grauert–Riemenschneidervanishing theorem (due to Huckaba and Huneke) to ideals satisfyingthe second Valabrega–Valla condition. The positivenessof Hilbert coefficients is also studied, and Valabrega–Vallaconditions are linked to the vanishing of the cohomology groupsof the closed fiber of the blowing up of Spec(R) along the closedsub-scheme defined by I.     

Gr-凝聚环的分次FP-内射维数

引进了分次FP-内射维数，对Gr-凝聚环的分次FP-内射维数作了刻划，将Stentrom等人的若干工作推广到分次环上．     

群分次环的分次Levitzki根

建立了分次环的分次Levitzki根,并给出了分次Levitzki半单环的结构定理及相关结果.     

V-Categories: Applications to Graded Rings

Motivated by the study of V-rings, we introduce the concept of V-category, as a Grothendieck category with the property that any simple object is injective. We present basic properties of V-categories, and we study this concept in the special case of locally finitely generated categories, for instance the category R-gr of all graded left R-modules, where R is a graded ring. We use the characterizations of V-categories in the study of graded V-rings. Since V-rings are closely related to Von Neumann regular rings (in the commutative case these classes of rings coincide), the last part of the article is devoted to graded regular rings.