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。     

分次Morita Contexts与分次根

设M={A=⊕_(g∈G)A_g,V=⊕_(g∈G)V_g,W=⊕_(g∈G)W_g,B=⊕_(g∈G)B_g}与(,),[,]是一个G-分次Morita Context,且满足(V,W)=A,[W,V]=B,A,B都有单位元．本文证明τG(B)：[W,ΥG(A)V]=【WΥc(A),V],ΥG(A)=(V,ΥG(B)W)=(VΥG(B),W)其中ΥG代表P_G(分次素根),J_G(分次Jacobson根),K_G(分次Koethe根),L_G(分次Levitzki根)和s_G(分次强素根),us_G(分次一致强素根)．     

分次Abel正则环的结构

首先给出了 gr-正则环为分次除环的两个充要条件 ,其次讨论了分次正则环r G(R)和分次 Jacobson根 JG(R)之间的关系 ,最后给出了分次 Abel正则环的结构定理 .     

分次根、弱分次根与自反根

本文在一般群分次环范畴中,给出了典型分次根NG,LG,gG,rG和KG的元素刻划.通过建立弱分次根概念,研究了自反根Nref,Lref,gref,rref和Kref的性质,探讨了这些分次根和与它们相对应的自反根的关系．     

G——可迁空间与分次本原环

本文给出分次环同构于Ｇ－－可迁空间上分次线性变换完全环的充要条件并且讨论了分次一致本原环的结构。     

分次P-根

在一般Monoid分次环范畴中定义了一种新的分次根——分次P-根,得到分次环的一个结构定理,证明分次P-根是一个分次特殊根,给出了它的分次模刻划,讨论了它与自反P-根的关系。     

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.     

弱分次周期根

对一般群分次环境立了弱分次周期根的概念,证明它是一个分次特殊根,给出了它的分次模刻,讲座了它与自反周期根的关系。     

分次环的有限正规扩张

本文研究了群分次环的有限正规分次扩张问题．利用经典环论方法，得到一个群分次环与其有限正规分次扩张环之间关于分次Jacobson根和分次素根的关系，同时，给出了分次情形的Cutting down定理和Lying over定理．     

分次环的分次σ-根

在分次环类中定义H-关系,利用H-关系给出分次σ-根的定义,并进一步得到分次环类γ是分次σ-根的几个充分必要条件.     

Graded Antisimple Primitive Radical

We introduce the graded version of the antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}} $ , the graded antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}}_{G} $ . We show that $ {\user1{\mathcal{S}\mathcal{J}}}_{G} = {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} = {\user1{\mathcal{S}\mathcal{J}}}^{G} $ when |G| < ∞, where $ {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} $ denotes the reflected antisimple primitive radical and $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ denotes the restricted antisimple primitive radical. Furthermore, we discuss the graded supplementing radical of $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ .     

Graded Antisimple Radicals

We introduce the graded version of the antisimple radical S, the graded antisimple radical SG. When |G| < , we show that SG = Sref = SG, where Sref denotes the reflected antisimple radical and SG denotes the restricted antisimple radical.     

ON ANTISIMPLE GAMMA RINGS

ABSTRACT

In this note, we define the antisimple radical, A(M), of a Γ-ring M. A(M) is shown to be a special radical, and two characterizations of antisimple rings due to Szész are extended to Γ-rings. If R is the right operator ring of M, then A(R)* = A(M), where A(R) is the antisimple radical of R. If m,n are positive integers, then A(M_mn) = (A(M))_mn, where M_mn denotes the group m x n matrices over M, considered as a Γ_nm -ring with the operations of matrix addition and multiplication.     

On graded Thierrin radicals of graded rings

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring.     

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．     

关于等差数列型素数集的分解

设C为非负整数集.若存在非负整数集A和B,使得C=A+B,其中|A|,|B|≥2,则称C为可分解的,否则称C为不可分解的或本原的.讨论特殊的等差数列型素数集合,证明了4k+1型素数集不可分解,以及4k+3型素数集不可分解.     

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.