Nilpotent Elements and Nil-Reflexive Property of Generalized Power Series Rings

Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

Principal Quasi-Baerness of Rings of Generalized Power Series

Let R be a ring such that all left semicentral idempotents are central and （S, ≤） a strictly totally ordered monoid satisfying that 0 ≤s for all s ∈S. It is shown that [[R^S≤]], the ring of generalized power series with coefficients in R and exponents in S, is right p.q.Baer if and only if R is right p.q.Baer and any S-indexed subset of I（R） has a generalized join in I（R）, where I（R） is the set of all idempotents of R.

PS-modules over Rings of Generalized Power Series

Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.

Principal Quasi-Baerness of Rings of Skew Generalized Power Series

Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,（R） has a generalized join in S,（R）. Several known results follow as consequences of our results.

多级谐振充电的Marx脉冲源研究

Marx发生器是一种经典的脉冲功率发生装置。为了更好地满足高电压、双极性高效率等实际使用需求,文中提出了多级谐振充电的Marx脉冲源,即利用串联谐振充电源与串心磁环变压器对Marx脉冲发生器分组充电。搭建两类对称结构多级谐振充电的Marx脉冲发生器并验证其性能,包括可变极性的单极性发生器和双极性脉冲发生器。电路由一组正极性的3路8级Marx和一组负极性的3路8级Marx组成,共48级。实验结果表明,可变极性的单极性Marx发生器可产生正极性或负极性电压幅值为30 kV的脉冲,双极性发生器可向空载和50 kΩ的阻性负载输出±10 kV的电压脉冲。通过FPGA(Field Program Gate Array)的信号控制可以将重复频率控制在1 Hz~1 kHz范围内调节,脉宽可在1~200μs范围内调节,脉冲前后沿为50 ns以内。正、负脉冲之间的弛豫时间为0.2~200.0μs。

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].

Morita Duality for the Rings of Generalized Power Series

Let A,B be associative rings with identity,and(S.≤)a strictly totally ordered monoid which is also artinian and finitely generated.For any bimodule AaMB. we show that the bimodule [[A^(S.≤)]][M^(S.≤)][[B^(S.≤)]]defines a Morita duality if and only if _AM_B defines a Morita duality and A is left noetherian.B is right noetherian.As a corollary,it.is shown that the ring[[A^(S.≤)]]of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _AM_B such that B is right noetherian.

PP-Rings of Generalized Power Series

As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and （S.≤） a strictly totally ordered monoid. We prove that （1） the ring [[R(S.≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B（R） （the set of all idempotents of R） has a least upper bound in B（R）. and （2） if （S. ≤） also satisfies the condition that 0≤s for any s∈S, then the ring [[R(S.≤]] is weakly PP if and only if R is weakly PP.

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

A generalization of semiprime rings and right p.q.-Baer rings,which we call quasi-Armendariz rings of differential inverse power series type(or simply,DTPS-quasi-Armendariz),is introduced and studied.It is shown that the DTPS-quasi-Armendariz rings are closed under direct sums,upper triangular matrix rings,full matrix rings and Morita invariance.Various classes of non-semiprime DTPS-quasi-Armendariz rings are provided,and a number of properties of this generalization are established.Some characterizations for the differential inverse power series ring R[[x^-1;δ]]to be quasi-Baer,generalized quasi-Baer,primary,nilary,reflexive,ideal-symmetric and left AIP are conncluded,whereδis a derivation on the ring R.Finally,miscellaneous examples to illustrate and delimit the theory are given.

广义逆多项式模的相伴素理想和素子模

讨论了广义幂级数环 [[RS,≤ ]]上的广义逆多项式模 M[S]的相伴素理想和素子模 ,指出它们与环R上模

一般Clean环的扩张

有单位元的结合环R称为clean环,如果R中的每个元素都可以写成一个幂等元与一个单位的和的形式.W.K.Nicholson,Y.Zhou[1]给出了一般clean环的定义.在此基础上讨论了一般clean环的几个扩张性质,得到如下结论:1)一般环I是一般clean环当且仅当I的形式幂级数环是一般clean环当且仅当I的斜幂级数环是一般clean环.2)若I是一般clean环,则对任意n≥1,均有I[x]/<xn+1>是一般clean环.

Meta-sided exchange环及其扩张

讨论了meta-sided exchange环的性质。证明了如果R是Abelian meta-sided exchange环,则对R的任意素理想P,都有R/P是局部环;如果R是Abelian环,(S,≤)是严格序幺半群且对任意s∈S,都有0≤s,则广义幂级数环[[RS,≤]]是meta-sided exchange环当且仅当R是meta-si-ded exchange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.

NCI环的多项式环未必是NCI环

环R指的是结合环但未必含有单位元.环R称为NCI环如果当它的幂零元集合N(R)≠0时那么N(R)包含R的一个非零理想.主要研究有关NCI环的性质,证明了存在NCI环R但是R的多项式环R[x]非NCI环,这否定地回答了S.U.Hwang等人(Bull.Korean Math.Soc.44(2007),No.2)的一个公开问题.进一步证明了如果环R的多项式环R[x]是NCI环则R是NCI环.此外还证明了存在NCI环但它的幂级数环不是NCI环,而如果环R的幂级数环为R[[x]]是NCI环那么R是NCI环.最后证明了如果环R存在非零的局部幂零理想I那么R的全矩阵环Mn(R)是NCI环.

一类广义幂级数环的有限维数

令R是一个有单位元的完备的凝聚交换环,研究并比较了R的有限维数与R上的广义幂级数环[[R≤,S]]的有限维数的关系,得到了一些有限维数不等式.结果表明:如果R是一个完备的凝聚的有单位元的交换环,则R的有限投射维数不超过[[RS,≤]]的有限投射维数;令R是一个完备的凝聚的有单位元的交换环,则R的有限内射维数不超过[[RS,≤]]的有限内射维数;如果R是一个完备的凝聚的有单位元的交换环,则R的有限弱维数不超过[[RS,≤]]的有限弱维数.

Semiclean环及其扩张

对clean环进行了推广,研究了semiclean环,讨论了semiclean环的几个重要性质;证明了(1)R是semi-clean环;(2)R上的形式幂级数环R[[x]]是semiclean环;(3)R上的斜幂级数环R[[x;α]]是semiclean环等价;最后证明了,如果R是环,(S,≤)是严格偏序幺半群且对任意s∈S,都有0≤s,则[[RS,≤]]是semiclean环当且仅当R是semiclean环.

π-正则广义幂级数环

讨论广义幂级数环 [[RS,≤ ]]及其系数环 R关于正则性的一些关系问题 ,给出广义幂级数环为 π-正则的条件和两个推论 .

广义逆多项式模的Attached素理想

设R是有单位元1的结合环,(S,≤)是严格全序Artin幺半群,M_R是右R-模,Att(M_R)与Att([M^(S,≤)]_([[R^(S,≤)]]))分别表示模M_R与广义逆多项式模[M^(S,≤)]_([[R^(S,≤)]])的所有Attached素理想组成的集合.该文主要讨论了广义幂级数环[[R^(S;≤)]]广义逆多项式模[[R^(S;≤)]]的Attached素理想的相关性质,证明了在一定条件下,有Att([M^(S,≤)]_([[R^(S,≤)]])={[[PR^(S;≤)]]P∈Att(M_R)}.这一结论表明广义逆多项式模([M^(S,≤)]_([[R^(S,≤)]])的Attached素理想在一定条件下可以用模M_R的Attached素理想来刻画,推广了Annin S在文献[1]中关于斜多项式环上逆多项式模的Attached素理想的相关结论.

三类广义幂级数环及其K_0群(英文)

在这篇文章中,我们探讨了三类广义幂级数环,并且刻画了这三类广义幂级数环的K_0群.