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 .     

Generic Flatness and the Jacobson Conjecture

A result of Artin, Small, and Zhang is used to show that a Noetherian algebra over a commutative, Noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, Noetherian associated graded ring. This result is extended to show that if an algebra over a commutative Noetherian ring has a locally finite, Noetherian associated graded ring, then the intersection of the powers of the Jacobson radical is nilpotent. The proofs rely on a weak generalization of generic flatness and some observations about G-rings.     

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)．     

J-clean环的推广

如果R中每个元素(对应地,可逆元)均可表示为一个幂等元与环R的Jacobson根中一个元素之和,则称环R是J-clean环(对应地,UJ环).所有的J-clean环都是UJ环.作为UJ环的真推广,本文引入GUJ环的概念,研究GUJ环的基本性质和应用.进一步地,研究每个元素均可表示为一个幂等元与一个方幂属于环的Jacobson根的元素之和的环.     

MORPHIC MODULES

A module M is called morphic if M/M α ? ker(α) for all endomorphisms α in end(M), and a ring R is called a left morphic ring if _RR is a morphic module. We consider the open question when the matrix ring M_n(R) is left morphic by relating it to when Rⁿ is morphic as a left R-module. More generally, we investigate when M being morphic implies that end(M) is left morphic, and conversely. Finally, we relate the morphic condition to internal cancellation in the module.     

Right Bézout ring with waist is a right Hermite ring

We study noncommutative rings in which the Jacobson radical contains a completely prime ideal. It is proved that a right Bézout ring in which the Jacobson radical contains a completely prime ideal is a right Hermite ring. We describe a new class of Bézout rings that are not elementary divisor rings.     

ON RELATIVE FLATNESS

Abstract

For a torsion radical, δ, we study various types of relative flatness and regularity. We obtain conditions valid when every R-module is δ-flat, when every R-module is semi-δ-flat and when every R-module is semi-δ-injective, and hence we characterize quasi-Frobenius rings R together with any torsion radical, δ, on R-mod. We define a ring to be δ perfect whenever every δ-flat module is projective and obtain extensions of some known results on perfect rings. We also introduce a relative form of the Jacobson Radical defined in terms of δ-flatness.     

所有同态象其Jacobson根与单根相同的Γ-环

设M是Nobusawa意义下的Г-环,S.Kyuno定义了环M_2=其中R,L分别是M的右、左算子环.本文首先刻画了环M_2的本原理想与Ja-cobson根.其次引进了一类新的Г-环称为PM Г-环,建立了Г-环M、矩阵Г_(n,m)-环M_(m,n)、Г-环M的右(左)算子环R(L)、M-环Г及M_2的PM性质之间的关系.最后,给出了Г-环一般形式的Jacobson性质,Jacobson性质、Brown-McCoy性质以及PM性质为其特殊情况.     

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．     

J-半单亚直既约环类确定的上限

Ｆ．Ａ．Ｓｚａｓｚ在［１］中提出公开问题５５：设Ｋ是Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类,研究类Ｋ确定的上根．本文对此进行了研究,证明了Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类Ｋ确定的上根Ｒ是特殊根,它介于Ｊａｃｏｂｓｏｎ根与Ｂｒｏｗｎ－ＭｃＣｏｙ根之间．并给出任意结合环Ａ为Ｒ－根环的充要条件．     

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.     

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.     

On ℵ0-Self-Injective Rings Modulo Their Jacobson Radical #

A well-known result by Y. Utumi states: Every right or left self-injective ring is von Neumann regular modulo its Jacobson radical. In this note, we give an example of a commutative ?₀-self-injective ring which is not von Neumann regular modulo its Jacobson radical.     

Strongly Gorenstein graded modules

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.     

The topological jacobson radical of rings. II

In this paper, topologically primitive rings and rings possessing a faithful topologically irreducible module and bounded by this module are considered for the investigation of properties of their topological Jacobson radical. We investigate the topological Jacobson radical in some classes of topological rings such as left topologically Artinian rings, topological rings possessing a basis of neighborhoods of zero consisting of ideals, compact rings, and bounded strictly linearly compact rings.     

On the structure of parabolic subgroups

The main aim of this paper is to show that the Jacobson and Brown-McCoy radicals of rings graded by free groups are homogeneous. As an application we get some information on the structure of the Jacobson radical of monomial rings. In particular we give a positive answer to a question posed in [12]. We extend also a result of [13] on the Brown-McCoy radical of polynomial rings in non-commutative variables. Actually this and the question of [12] motivated our studies.     

Jacobson rings and rings strongly graded by an abelian group

LetR be ring strongly graded by an abelian groupG of finite torsion-free rank. Lete be the identity ofG, andR _e the component of degreee ofR. AssumeR _e is a Jacobson ring. We prove that graded subrings ofR are again Jacobson rings if eitherR _e is a left Noetherian ring orR is a group ring. In particular we generalise Goldie and Michlers’s result on Jacobson polycyclic group rings, and Gilmer’s result on Jacobson commutative semigroup rings of finite torsion-free rank.     

PRIME AND RADICAL SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

ABSTRACT

In this paper, unless otherwise stated, all rings are commutative with identity and all modules are unital. We give sufficient conditions to ensure that a submodule has a module-reduced primary decomposition. In general, the radical of a primary submodule is not prime and the radical does not split intersections of submodules, as is valid in the ideal case. We study sufficient conditions for which these properties hold in the module setting. These conditions involve dimension arguments, consideration of finitely generated modules, and the spectrum of a given prime ideal. Further, we consider the computational problem of finding a Gröbner basis of both the colon and the radical of a submodule. A characterization of the elements of the colon is given, along with a method of computing the radical of a submodule in certain cases.     

关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.