P—内射环和半素环

本文主要证明了如下结果:1 如果 R 是左 p-环,那未(a)Z(R)=J(R);(b)若 R 的每个非零左理想包含极小左理想,则 J(R)=r(Socle_RR)。2 如果 R 是半素的左 p-环,那未(a)R 有唯一的最大理想 I,I 不含非零幂零元,且I=lr(I)=rl(I),Z(_RI)=Z(I_R)=0,(b)R 有极大左零化子当且仅当 Socle R≠0.     

半素环的(1,σ)导子扩张

本文给出了半素环的自同构和（ １,σ）导子的扩张命题,这推广了［１］中的一些结论,然后借助半素环的广义中心里的幂等元在（１,σ）导子扩张模上建立起拓扑空间,从拓扑学角度给出（１,σ）导子扩张模的结构定理．     

半素环上(右)广义自同构的扩张

In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.     

半素环的一个交换性条件

设R是一个半素环,Z(R)的R的中心,本证明了：如果对任意：x,y∈Z(R),那么,R是一个交换环。     

具某些有限条件的半素环

设Ｒ是环，Ｃ（Ｒ）＝｛ｘ｜ｘＲ＝Ｒｘ，ｘ∈Ｒ｝。本文证明了对于半素ＥＳ－环Ｒ，若Ｃ（Ｒ）中仅有有限个非零幂等元，则下列条件等价：（１）Ｒ只有有限个非零幂等元∈Ｒ－Ｃ（Ｒ）。（２）Ｒ只有有限个非零幂零元。（３）Ｒ只有有限个非零元ｘ：ｘ２＝０。（４）Ｒ同构于有限个除环或有限域上有限阶全矩阵环（阶数至少大于２，个数至少大于１）的直和     

关于拟P-内射模的一些注记(英文)

设 R是一个环 .一个右 R-模 M叫做拟 P-内射的 ,如果 M的每个 M-循环子模到 M的任一个 R-同态都能扩展到 M.假设 M是一个自生成子的拟 P-内射模 .在这篇文章中 ,我们表明如果这样一个模是一个 CF-模 (特别地 ,CS-模 ) ,那么 S/J(S)是正则的 ,其中 S=End(MR) .进一步 ,如果 S是半素环 ,那么 M的每个极大核是 M的一个直和项 .这些结果扩展了 P-内射环的一些结果     

半模弱Brandt半群（英文）

全子半群定义为包含所有幂等元的子半群.众所周知,一个半群所有全子半群关于集合的包含关系构成格.一个ample半群称为分配的(模的;半模的),如果其全子半群格为分配格(模格;半模格).本文得到了弱Brandt半群成为半模(模;分配)ample半群的充分必要条件.作为应用,确定了本原半单ample半群何时为模(分配)ample半群.     

Zip模的扩张（英文）

本文主要证明了:(1)如果右R-模MR是(α,δ)-compatible且(α,δ)-Armendariz,则右R[x;α,δ]-模M[x]是zip模当且仅当右R-模MR是zip模;(2)如果(S,)是可消无挠严格序幺半群且M_R是S-Armendariz模,则右[[R~S,]]-模[[M~S,]]_([[R~S,]]是zip模当且仅当右R-模M_R是zip模;(3)如果M_R是reduced且σ-compatible模,G为序群,则Malcev-Neumann环R*((G))上模M*((G))_(R*((G)))是zip模当且仅当右R-模M_R是zip模;因此一些文献中关于zip环与zip模的部分结论可以看作是本论文相关结论的推论.     

微商共同作用在半素环的Lie理想上的结果

本文讨论了微商共同作用在半素环的某个Ｌｉｅ理想上的问题。给出了如下结果：设Ｒ是带有中心Ｚ（Ｒ）的半素环,Ｑｍｒ是Ｒ的极大右商环,Ｌ是Ｒ的非交换Ｌｉｅ理想,ｄ和δ是Ｒ的微商,假设ｒＲ（「Ｌ,Ｌ」）＝０且ｄ（ｘ）ｘ－ｘδ（ｘ）∈Ｚ（Ｒ）对任意ｘ∈Ｌ成立,则在Ｒ的扩张形心Ｃ中存在一个幂等元ｅ使得ｄ（１－ｅ）Ｑｍｒ＝０和δ（１－ｅ）Ｑｍｒ）＝０并且ｅＱｍｒ满足Ｓ４。另外给出微商共同作用在半素环上多项式的结     

Localization of Injective Modules Over Arithmetical Rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R _P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective.     

On Semiprime Goldie Modules

In this article, we investigate some properties of right core inverses. Particularly, new characterizations and expressions for right core inverses are given, using projections and {1, 3}-inverses. Also, we introduced and investigated a new generalized right core inverse which is called right pseudo core inverse. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses, and EP elements.     

Cyclically Presented Modules Over Rings of Finite Type

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules.     

Weak Dimension of FP-injective Modules Over Chain Rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Finitely Presented Modules Over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.     

n- Gorenstein环上的 Gorenstein内射模(英文)

用 Gorenstein内射模刻画了 n-Gorenstein环 .     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

On Semiprime Multiplication Modules over Pullback Rings

We classify all those indecomposable semiprime multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [9 Ebrahimi Atani , S. , Farzalipour , F. ( 2009 ). Weak multiplication modules over a pullback of Dedekind domains . Colloquium Math. 114 : 99 – 112 .[Crossref] , [Google Scholar]] to a more general semiprime multiplication modules case.     

FI性质的传递

一个环R称为左(右)FI环，如果它的每一个平坦左(右)R模是内射的，R称为FI环是指它既是左且右的FI环．本讨论了当R是FI环时，其多项式环R[t]，矩阵环MN(R)以及分式不S^-1R也是FI环的充分与必要条件．