迭代的斜多项式环的Baer和拟-Baer性

本文主要讨论了环R和迭代的斜多项式环T(u)的零化子之间的关系,从而得出在一定条件下,R是Baer环当且仅当T(u)是Baer环。而对于拟-Baer性,只要R是拟Baer环就行了,作为推论我们证明了sl(2)的包络代数和量子包络代数都是拟Baer环。     

Malcev-Neumann环的主拟Baer性质

设R是环，G是偏序群，σ是从G到R的自同构群的映射。本文研究了Malcev-Neumann环R*((G))是主拟Baer环的条件。证明了如下结果：如果R是约化环并且σ是弱刚性的，则R*((G))是主拟Baer环当且仅当R是主拟Baer环，并且I(R)的任意G可标子集在I(R)中具有广义并．     

广义幂级数环的拟Baer性

设R是环,（S,≤）是严格全序幺半群,且对任意s∈S都有0≤s.本文证明了环R是拟Baer环当且仅当R上的广义幂级数环[[RS,≤]]是拟 Baer环.     

广义幂级数环的拟Baer性

设R是环,(S,≤)是严格全序幺半群,且对任意s∈S都有0≤s.本文证明了环R是拟Baer环当且仅当R上的广义幂级数环[RS,≤]]是拟Baer环。     

斜幂级数环的主拟Baer性

设R是环,并且R的左半中心幂等元都是中心幂等元, α是R的一个弱刚性自同态. 本文证明了斜幂级数环R[[x,α]]是右主拟Baer环当且仅当R是右主拟Baer环,并且R的任意可数幂等元集在I(R)中有广义交,其中I(R)是R的幂等元集.     

罗朗级数环的主拟Baer性

称环 R为右主拟 Baer环（简称为右p·q.Baer环）,如果 R的任意主右理想的右零化子可由幂等元生成.本文证明了,若环 R满足条件Sl（R）（?）C（R）,则罗朗级数环R[[x,x-1]]是右p.q.Baer环当且仅当R是右p.q.Baer环且R的任意可数多个幂等元在I（R）中有广义join.同时还证明了,R是右p.q.Baer环当且仅当R[x,x-1]是右P.q.Baer环.     

关于有限生模嵌入自由模

本文利用模的H-有限生成性质刻画了具有性质任意有限生成左模是自由模的子模的环.另外,还给出了左IF-环的一个刻画.     

拟内射模中的消去定理

当左拟内射模Ｍ的自同态环Ｅｎｄ_RＭ为一Ｄｅｃｋｉｎｄ有限环时．Ｍ的任何两个相互同构的子模的左相关补子横也同构。     

$FI$-$t$-提升模和$t$-quasi-dual Baer模

本文引入$FI$-$t$-提升模和$t$-quasi-dual Baer模的概念并给出两者的联系.证明富足补模$M$为$FI$-$t$-提升模当且仅当$M$的每个完全不变$t$-coclosed子模为$M$的直和项当且仅当$\bar{Z}^{2}(M)$为$M$的直和项且$\bar{Z}^{2}(M)$为$FI$-$t$-提升模当且仅当$M$同时为$t$-quasi-dual Baer 模和$FI$-$t$-$\mathcal{K}$-模.     

纯拟内射模

本文引进了纯拟内射模的概念，讨论了该模的一些主要性质，证明了纯拟内射模保持有限直和，进一步地利用这类新模刻画了正则环的特征。     

Principally Quasi-Baer Skew Power Series Modules

A module M _R is called principally quasi-Baer (or simply p.q.-Baer) if the annihilator of every cyclic submodule of M _R is generated by an idempotent, as a right ideal. Let α be an automorphism of R and M _R be an α-compatible module and every countable subset of right semicentral idempotents in R has a generalized countable join or R satisfies the ACC on left annihilator ideals. It is shown that M _R is p.q.-Baer if and only if M[[x]] _{R[[x; α]]} is p.q.-Baer if and only if M[[x, x ^?1]] _{R[[x, x ^?1; α]]} is p.q.-Baer. As a consequence, we unify and extend nontrivially many of the previously known results such as [11 Hashemi , E. ( 2008 ). A note on p.q.-Baer modules . New York J. Math. 14 : 403 – 410 . [Google Scholar], 15 Huang , F. K. ( 2008 ). A note on extensions of principally quasi-Baer rings . Taiwan. J. Math. 45 ( 4 ): 469 – 481 . [Google Scholar], 20 Liu , Z. ( 2002 ). A note on principally quasi-Baer rings . Comm. Algebra 30 ( 8 ): 3885 – 3890 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Examples to illustrate and delimit the theory are provided.     

关于主伪内射模

对于环R．一个右R模被叫做主伪内射模。若每一个从M的主子模到M的单同态可以扩张为M的自同态．主伪内射是主拟内射的推广．在本文中,我们给出了一些主伪内射的性质并讨论什么情况下主伪内射模是主拟内射模的问题．     

Principally Quasi-Baer Skew Power Series Rings

Let α be an endomorphism of R which is not assumed to be surjective and R be α-compatible. It is shown that the skew power series ring R[[x; α]] is right p.q.-Baer if and only if the skew Laurent series ring R[[x, x ^?1; α]] is right p.q.-Baer if and only if R is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join. Examples to illustrate and delimit the theory are provided.     

伪内射模与主伪内射模

本文研究了伪内射模与主伪内射模,它们分别是拟内射模与PQ-内射模的推广．伪内射模是对偶于伪投射模的．我们讨论了伪内射模与主伪内射模的性质及其自同态环,并得到了自同态环的Jacobson根的若干性质．     

拟-Armendariz模

For a right R-module N, we introduce the quasi-Armendariz modules which are a common generalization of the Armendariz modules and the quasi-Armendariz rings, and investigate their properties. Moreover, we prove that NR is quasi-Armendariz if and only if Mm(N)Mm(R) is quasi-Armendariz if and only if Tm(N)Tm(R) is quasi-Armendariz, where Mm(N) and Tm(N) denote the m×m full matrix and the m×m upper triangular matrix over N, respectively. NR is quasi-Armendariz if and only if N[x]R[x] is quasi-Armendariz. It is shown that every quasi-Baer module is quasi-Armendariz module.     

A note on semicentral idempotents

In this note we answer the question raised by Han et al. in [3 Han, J., Lee, Y., Park, S. (2014). Semicentral idempotents in a ring. J. Korean Math. Soc. 51(3):463–472, MR3206399.[Crossref], [Web of Science ®] , [Google Scholar]] whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.     

On a class of semicommutative modules

Let R be a ring with identity, M a right R-module and S = End _R (M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module.     

Prime ideals of principally quasi-Baer rings

We extend a theorem of Kist for commutative PP rings to principally quasi-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Also decompositions of quasi-Baer and principally quasi-Baer rings are investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M _R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M _R is p.q.-Baer if and only if the generalized power series module M[[S]] _{R[[S, ω]]} is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.     

Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Abstract

For a ring S, let K ₀(FGFl(S)) and K ₀(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K ₀(FGFl(R)) → K ₀(FGFl(R/J(R))) is an epimorphism and K ₀(FGFl(R)) = K ₀(FGPr(R)).