半素环上的微商

本文讨论了微商作用在半素环的某些左理想上的问题.给出了如下结果:设R是带有中心Z（R）的半素环.d和g是R的微商,L为R的非零左理想且r_R（L）=0.假设d（x）x-xg（x）∈Z（R）对任意的x∈L成立.那么d（R）?Z(R）且由d（R）生成的R的理想在R的中心里.     

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     

关于半素环上微分恒等式的某些结果

本文主要讨论了半素环的微商满足某些分恒等式的问题,给出了如下结果：假设R是带有扩张形心C的素环,I为R的稠密单侧理想,d和g是Ｒ的微商,如果d(x)g(x)=g(x)d(x)对任意x∈Ｉ均成立,那么对第一个x∈Ｒ,d^2(x)=0,或者存在λx∈Ｃ使得d(x)-λxg(x)∈Ｃ,同时给出了半素环的微商满足某些微分恒等式的结果。     

环的素谱的若干注记

对于可换环Ｒ,本通过考察ＳｐｅｃＲ和Ｈ０Ｒ,得到了关于广义素环和广义半素环的一些结果,另一方面,本还得到了有关连通性与道路连通性的几个结果。     

半素环上的右理想及其微商

R是半素环,d是R的微商,ρ是R的右理想,a是R中元素,如果对于ρ中的所有元素x,都有ad(x)ⁿ=0,其中n是一个固定的正整数,那么必有aρd(P)P=0.     

半素环的一个交换性条件

本证明了以下定理：一个半素环是交换的当且仅当以下条件之一成立：(1)[x^my^n xy^nx,x]=0，(2)[x^sy^t yx^s,x]=0．其中x，y为R的任意元，m,n，s，t为正整数。     

半素环扩张中心上的模（英文）

设C为半素环R的扩张中心，通过C中的幂等元，我们可以在任意C－模上建立拓扑空间，从拓扑学角度出发，我们证明，殆Hausdorff C- 模为内射模，此外，我们给出殆Hausdorff C-模的一种有趣刻画：若M和N都是殆Hausdorff C-模，则存在一个幂等元e∈C，使得Me可嵌入到Ne中且N（1－e)可嵌入到M（1-e)中。     

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。     

关于环的变换性条件的若干注记

关于环的变换性条件的若干注记戴跃进（福建师范大学数学系，福州３５０００７）关键词字，半素环，Ｊ－半单环，次直和。分类号ＡＭＳ（１９９１）１６Ｕ８０／ＣＣＬＯ１５３．３本文运用下列熟知的引理证明了某些环的变换性定理并推广了Ｈ．Ｅ．Ｂｅｌｌ￣［４］中若干...     

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．     

ABSOLUTELY E-PURE MODULES AND E-PURE SPLIT MODULES

We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.     

PRIME AND SEMIPRIME BI-IDEALS

Abstract

Prime and semiprime bi-ideals in associative rings are defined. This provides a setting for a generalization of the well-known theorem that a commutative ring is Von Neumann regular iff every ideal is semiprime.     

HIGHER DERIVATIONS OF SEMIPRIME RINGS

ABSTRACT

In this paper we study higher derivations of prime and semiprime rings satisfying linear relations. We extend several results known for algebraic derivations, and we prove some other results.     

The radical-annihilator monoid of a ring

Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Nonadditive Strong Commutativity Preserving Maps

The goal of this note is to characterize strong commutativity preserving maps on prime or semiprime rings without the additivity assumption on the maps considered in the literature.     

CLOSED SUBMODULES OF NORMALIZING BIMODULES OVER SEMIPRIME RINGS

In this paper we study closed sub-bimodules of normalizing bimodules over semiprime rings. We extend the main results which are known for centred bimodules and several other results which are new even for centred bimodules are also obtained. In particular, we prove that the theorem on one-to-one correspondence between closed submodules obtained in former papers for centred bimodules is also true for normalizing bimodules. Finally, we give some applications of the main results.

     

RINGS WHOSE SIMPLE MODULES ARE ABSOLUTELY PURE

A ring is called right SAP if every right simple module over it is absolutely pure. In this paper we prove that every right SAP ring is semiprimitive and that the homomorphic image and the center of an right SAP ring are also right SAP. We also show that the sum of all absolutely pure minimal submodules of any module is a fully invariant submodule. As an application, we give a decomposition of some selfinjective rings.     

Semiprime Ideals and Separation Theorems for Posets

The concept of a semiprime ideal in a poset is introduced. The relations between the semiprime (prime) ideals of a poset and the ideals of the set of all ideals of the poset are established. A result analogous to Separation Theorem is obtained in respect of semiprime ideals. Further, a generalization of Stone’s Separation Theorem for posets is obtained in respect of prime ideals. Some counterexamples are also given.