Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

分次可除模

对于Ｇ—分次环Ｒ，我们证明如下结论：（１）若Ｒ是分次正则环，则Ｒ上的任一分次左Ｒ—模都是分次可除模；（２）若Ｒ分次非退化且Ｍ是分次可除左Ｒ—模，则Ｍｅ是可除左Ｒｅ—模；（３）若Ｇ是有序群，Ｍ是可除左Ｒ—模，则Ｍ～和Ｍ～是分次可除左Ｒ—模，其中Ｍ为分次左Ｒ—模Ｎ的子模     

Jacobson Radicals Graded by G-sets

ＪａｃｏｂｓｏｎＲａｄｉｃａｌｓＧｒａｄｅｄｂｙＧ－ｓｅｔｓＳｕｎＪｉａｎｈｕａ（孙建华）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｔｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，ＹａｎｇｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｙａｎｇｚｈｏｕ，Ｊｉａｎｇｓｕ，２２...     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

In this article, considering the difference between the finiteness dimension and cohomological dimension of a finitely generated graded module, we investigate the asymptotic behaviour of grades of components of a graded local cohomology module with respect to the irrelevant ideal. To this end, we study some Artinian and tameness properties of certain graded local cohomology modules.     

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.     

Generalized Green Correspondence of Graded Modules

The author studies the Green correspondence and quasi-Green correspondence for indecomposable modules over strongly graded rings. The motivation is to investigate the influence of induction and restriction processes on indecomposability of graded modules.     

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.     

使用相关内射性对拟连续模的一个刻划

近二十年，许多环与模工作对拟投射模与拟内射模作了各种推广与研究。连续模与拟连续模就是拟内射模的一种推广，拟连续模要比连续模弱。在[2]，作对连续模与拟连续模做了深入的研究。在这篇章中，利用相关内射性给出了拟连续模的一个刻划。     

A Note on the Graded K-Theory of Certain Graded Rings

Following ideas of Quillen we prove that the graded K-theory of a ? ⁿ -graded ring with support contained in a pointed cone is entirely determined by the K-theory of the subring of degree 0 elements.     

F－V－环的广义内射性刻划

设Ｆ是含单位元的结合环Ｒ上的左Ｇａｂｒｉｅｌ拓朴，称Ｒ是Ｆ－Ｖ－环，如果商范畴（Ｒ，Ｆ）－Ｍｏｄ中的所有单对象都是内射对象。本文我们利用左Ｒ－模的ｖＮ－内射性及拟内射性给出Ｆ－Ｖ－环的特征刻划。     

F－V－环的广义内射性刻划

设Ｆ是含单位元的结合环Ｒ上的左Ｇａｂｒｉｅｌ拓朴，称Ｒ是Ｆ－Ｖ－环，如果商范畴（Ｒ，Ｆ）－Ｍｏｄ中的所有单对象都是内射对象。本文我们利用左Ｒ－模的ｖＮ－内射性及拟内射性给出Ｆ－Ｖ－环的特征刻划。     

Injectivity of directed complete poset acts

Action of a monoid on a set have always been of interest for mathematicians and computer scientists. On the other hand, Domain Theory, which studies directed complete partially ordered sets, was introduced by Scott in 1970 when he explored the possibility of using ordered topological spaces to give meaning first to typed and then untyped λ-calculi.

In this paper, combining the above two notions, we consider actions of a monoid on domains and study the algebraic notion of injectivity with respect to monomorphisms and embeddings in the category so obtained.     

On Pseudo Small Injective Modules

The definition of principally pseudo injectivity motivates us to generalize tae notion of injectivity, noted SP pseudo injectivity. The aim of this paper is to investigate characterizations and properties of SP pseudo injective modules. Various results are devel- oped, many extending known results. As applications, we give some characterizations on Noetherian rings, QI rings, quasi-Frobenius rings.     

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．     

分次反单本原根

本文给出分次情形的反单本原根SJGG,当群G满足|G|<∞,证明了SJG=SJref=SJG,其中SJref和SJG分别表示反射反单本原根和限制反单本原根.而且讨论了SJG的分次补根.     

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} $ .     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.