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.     

广义导子幂中心化子条件

设R是半素环,U是R的极大右商环,C是R的扩展型心,a,b∈U．对R上广义导子δ的值的幂零,幂中心化子问题进行了讨论,即(aδ(x))~nb=0(∈C)形式的恒等式,对δ的形式以及δ与a,b的关系做了完整的刻画,从而发展了Herstein,Brear的研究结果．     

On SZ°-Ideals in Polynomial Rings

The theory of z-ideals and z°-ideals, especially as pertaining to the ideal theory of C(X), the ring of continuous functions on a completely regular Hausdorff space X, has been attended to during the recent years; see Gillman and Jerison [9 Gillman , L. , Jerison , M. ( 1976 ). Rings of Continuous Functions . New York : Springer Verlag . [Google Scholar]], Mason [18 Mason , G. ( 1989 ). Prime ideals and quotient rings of reduced rings . Math. Japan 34 : 941 – 956 . [Google Scholar]], and Azarpanah et al. [4 Azarpanah , F. , Karamzadeh , O. A. S. , Rezaei Aliabad , A. ( 2000 ). On ideals consisting entirely of zerodivisor . Comm. Algebra 28 ( 2 ): 1061 – 1073 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In this article we will consider the theory of z°-ideals as applied to the rings of polynomials over a commutative ring with identity. We introduce and study sz°-ideals (an ideal I of a ring is called sz°-ideal, if whenever S is a finite subset of I, then the intersection of all minimal prime ideals containing S is in I). In addition, we will pay attention to several annihilator conditions and find some new results. Finally, we use the two examples that appeared in Henriksen and Jerison [10 Henriksen , M. , Jerison , M. ( 1965 ). The space of minimal prime ideals of a commutative ring . Trans. Amer. Math. Soc. 115 : 110 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]] and Huckaba [12 Huckaba , J. A. ( 1988 ). Commutative Rings with Zero Divisors . Marcel-Dekker Inc . [Google Scholar]], to answer some natural questions that might arise in the literature.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

Coherence and Generalized Morphic Property of Triangular Matrix Rings

Let R be a ring. R is left coherent if each of its finitely generated left ideals is finitely presented. R is called left generalized morphic if for every element a in R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ring T _n (R) (n ≥ 1). It is shown that R is left coherent if and only if T _n (R) is left coherent for each n ≥ 1 if and only if T _n (R) is left coherent for some n ≥ 1. And an equivalent condition is obtained for T _n (R) to be left generalized morphic. Moreover, it is proved that R is left coherent and left Bézout if and only if T _n (R) is left generalized morphic for each n ≥ 1.     

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.     

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,≤]].     

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.     

A Relative Filtration Index and Fibers of Normal Primes in Extensions of Finite Type

For an extension R A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the 'splitting' (or 'branching') and the 'ramification' that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a 'normal point' of Spec R, that is, over p Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be 'unibranched', i.e., the Henselization of R/p is a (normal) domain].     

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.     

关于wF(p,r,q)算子类

作为wA(p,r)算子类的一个推广,该文介绍了一类更广泛的算子类即 wF(p,r,q)算子类,它包含A(p,r)类而含于F(p,r,q)类之中, 进而考虑了该类算子的特征, 包含关系, 正规性和幂性质等等.