关于强Armendariz环和强拟Armendariz环

提出了强拟Armendariz环的概念,给出了强Armendariz环和强拟Armendariz环上的一些结果.     

关于强P除环上方阵的酉相似理论的一点注记

在文[1]中定义了强p除环Ω,即满足如下条件(1)—(4)的除环Ω: (1)存在Ω的对合反自同构σ(即σ为反自同构,且σ(σ(α))=α Aα∈Ω) (2)Aα_i∈Ω,i=1,…,n(n∈N) sum from i=1 to n(α_iσ(α_i)=0 α_i=0,i=1,2,…,n)。 (3)命R={α∈Ω|σ(α)=α},则R含在Ω的中心中。 (4)Aα_i∈Ω,i=1,2,…,n(n∈N)方程x~2-sum from i-1 to n(α_iσ(α_i))=0在Ω中有且只有两解。 事实上,除了平凡的情况外,强p除环Ω就是R上的四元数除环。确切地说,我们有 定理1 设Ω为强p除环,则Ω为(1)R,(2)R+R_i或(3)R上的四元数除环。这里     

关于半交换环与强正则环

本文得到了环Ｒ是强正则环的若干充分必要条件，证明了下面条件是等价的：（１）Ｒ是强正则的；（２）Ｒ是半交换正则的；（３）Ｒ是半交换的左ＳＦ－环；（４）Ｒ是半交换的ＥＬＴ环，且使得每个单左Ｒ－模是Ｐ－内射的或者平坦的；（５）Ｒ是半交换右非奇异的左ＳＦ－环；（６）Ｒ是半素的半交换左（或右）Ｐ－内射环．     

置换环上的稳定矩阵

本文证明了置换环上的正则稳定矩阵是幂等矩阵和可逆矩阵的积,进一步证明了置换环上的正则稳定矩阵可以对角化。     

置换QB-环上的有限生成投射模

研究了置换QB-环上的有限生成投射模,证明了QB-环可以通过有限投射生成子的外替换和内替换来刻画.同时,还研究了QB-环上投射模的消去性.     

关于Z／mZ上置换多项式向量组的一个注记

设ｍ和ｎ是两个正整数，Ｒ＝Ｚ／ｍＺ是整数模ｍ剩余类环．设ｎ个多项式ｆｊ（ｘ１，…，ｘｎ）∈Ｒ［ｘ１，…，ｘｎ］，ｊ＝１，…，ｎ，若对任给的（ａ１，…，ａｎ）∈Ｒｎ，同余式组ｆｊ（ｘ１，…，ｘｎ）≡ａｊ（ｍｏｄｍ），ｊ＝１，…，ｎ，恰有一个解，则称ｆ１，…，ｆｎ是一组置换多项式向量模ｍ．一个自然的问题是：Ｒ上哪些ｎ元多项式可以扩充为一组置换多项式向量模ｍ？本文解决了这一问题，并给出了一个充分必要条件．     

关于GP-V-环与强正则环的刻划

The purpose of this paper is to characterize strongly regular rings via MERT rings and weakly one-sided ideals. Many important equivalent conditions on strongly regular rings are shown.     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

关于Buchsbaum环的注记

本文证明了下述结论，设Ａ是一个级数为ｄ的Ｂｕｃｈｓｂａｕｍ环，（ａ₁，ａ₂，…，ａ_n）是Ａ的一个参数系统，则任何正整数ｎ，Ａ／（ａ₁，ａ₂，…，ａ_k）ⁿ（１≤ｋ≤ｄ）仍是ｄ－ｋ维的Ｂｕｃｈｓｂａｕｍ环．     

A Note on Elements and Modules over Exchange Rings

ＬｅｔＲｂｅａｎａｓｓｏｃｉａｔｉｖｅｒｉｎｇｗｉｔｈｉｄｅｎｔｉｔｙ．ＡｒｉｇｈｔＲ－ｍｏｄｕｌｅＡｉｓｓａｉｄｔｏｓａｔｉｓｆｙｆｉｎｉｔｅｅｘｃｈａｎｇｅｐｒｏｐｅｒｔｙｉｆｆｏｒｅｖｅｒｙｒｉｇｈｔＲ－ｍｏｄｕｌｅＫａｎｄａｎｙｔｗｏｄｅｃｏ...     

A Note on Strongly Clean Matrix Rings

Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e ² = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7 Nicholson , W. K. ( 1999 ). Strongly clean rings and Fitting's lemma . Comm. Algebra 27 : 3583 – 3592 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring 𝕄 _n (R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 𝕄₂(RC ₂) over the group ring RC ₂ with R local is obtained.     

A Question on Strongly Clean Rings

A new characterization of a strongly clean ring is given. And it is proven that if R is a strongly clean ring, then eRe is a strongly clean ring for e ² = e ∈ R, which answers a question of Nicholson (1999 Nicholson , W. K. ( 1999 ). Strongly clean rings and Fitting's lemma . Comm. Algebra 27 ( 8 ): 3583 – 3592 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in the affirmative.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

关于Perfect环的一个注记

对于交换环R,Chase[1]证明:对任意集A,若R^A是射影模,则R是一个Artin环.而对非交换环,有例子说明,此结论不成立.本文讨论了对什么环,当R是射影模时,R是一个Artin环.     

关于自内射环的一个注记

设 R是含幺环 ,本文证明了在一定条件下 ,R与其一种有限单扩张同时具有自内射性 ,从而将欧海文等的主要结果定理 1推广到更大一类环上     

关于AP-内射环的一个注记

本文的主要目的是讨论AP-内射环中的两个问题:(1)环R是正则的当且仅当R是左AP-内射的左PP-环;(2)如果R是左AP-内射环,那么R是内射环当且仅当R是弱内射环.因此我们推广了内射环的一些结果,与此同时我们还取得了一些新的结果.