Poisson Prime Ideals of Poisson Polynomial Rings

Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ].     

Primitive and Poisson Spectra of Single-Eigenvalue Twists of Polynomial Algebras

We examine families of twists by an automorphism of the complex polynomial ring on n generators. The multiplication in the twisted algebra determines a Poisson structure on affine n-space. We demonstrate that if the automorphism has a single eigenvalue, then the primitive ideals in the twist are parameterized by the algebraic symplectic leaves associated to this Poisson structure. Furthermore, in this case all of the leaves are algebraic and can be realized as the orbits of an algebraic group. Presented by K. Goodearl     

Primitive Ideals in the Algebra of Regular Functions on Quantum m × n Matrices

Mathematical Notes -     

Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F _,p [G] depends on both a suitable root of unity and an antisymmetric bicharacter p: Z ⁿ ×Z ⁿ C ^×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F _,p [G] depends on a natural ×-action thereon, where is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (PrimF _,p [G])/× onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F _,p [G] is not finitely stratified by the torus action.     

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec _r (R) (resp. Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U _r (eR) = {P ? Spec _r (R)| e ? P}. Let  = ∪_{P?Prim r (R)} Spec _r ^P (R), where Spec _r ^P (R) = {Q ?Spec _r ^P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on  and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if  is a space with Zariski topology if and only if, for any Q ? , Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ? = Max _r (R) ∪  Prim _r (R) (resp.  = Prim _r (R)) there is an element e in R with e ² ? e ? J(R) such that U = U _r (eR) ∩  ? (resp. U = U _r (eR) ∩  ), where J(R) is the Jacobson of R. (3) Max _r (R) ∪  Prim _r (R) is connected if and only if Max _l (R) ∪  Prim _l (R) is connected if and only if Prim _r (R) is connected.     

Obstructions to the equivalence of Poisson structures near a symplectic leaf of semisimple and compact type

We describe cohomological obstructions to the equivalence of Poisson structures around a symplectic leaf of semisimple and compact type. The result is based on Conn’s linearization theorem and the theory of Poisson coupling.     

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。     

分次环的分次Jacobson根

本文通过引入弱拟正则元的概念，对一般Ｍｏｎｏｉｄ分次环Ａ（未必有１）给出以内部元素刻划的分次Ｊａｃｏｂｓｏｎ根ＪＧ（Ａ）．证明当Ａ有１时，ＪＧ（Ａ）与通常定义的Ｊｇ（Ａ）相等．对ＪＧ（Ａ）性质的讨论，推广了最近的许多结果．作为应用，我们给出了Ａｒｔｉｎ分次环的全部基本结构定理．     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

关于循环环的理想、素理想和极大理想

研究了循环环R=的理想、素理想和极大理想的个数和结构,得到了如下结论:1)理想:(1)若|R|=∞,则R共有无穷多个理想:;(2)若|R|=n,设n的正因数个数为T(n),则R共有T(n)个理想:.2)素理想:(1)若|R|=∞,设a^2=ka(k≥0),①当k=0时,R的素理想只有R;②当k>0时,R的素理想共有无穷多个,它们是:{0}、R及;(2)若|R|=n>1,设a^2=ka,0≤k.3)极大理想:(1)若|R|=∞,则R有无限多个极大理想,它们是;(2)若|R|=n>1,设n的互不相同的素因数个数为ψ(n),则R共有ψ(n)个极大理想:(pa|p是n的素因数).     

群分次环的本原性及分次本原性

设A是一个用有限群G分次的环,本文给出了Smash积A＃G~*为本原环的一个判别准则,并证明了群分次环的每一个本原理想必定包含一个分次本原理想。作为一个推论,得到已被Cohen和Montgomery证实的Bergman猜想的另一个证明。此外还得到了A_1为本原环或单环的判别。     

Linearizability of Poisson structures around singular symplectic leaves

The linearization problem for a Poisson structure near a singular symplectic leaf of nonzero dimension is studied. We obtain the following generalization of the Conn linearization theorem: if the transverse Lie algebra of the leaf is semisimple and compact, then the Poisson structure is linearizable, provided that certain cohomological obstructions vanish.     

关于nZ的理想及商环

给出了nZ的全部理想、极大理想和素理想,并研究了nZ的商环的构造以及为域的条件,解决了Z<,n>的子域的存在和个数问题.     

On Rings Whose Elements are the Sum of a Unit and a Root of a Fixed Polynomial

A ring R with identity is called “clean” if for every element a ? R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ? R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ? (x ? a)(x ? b)C(R)[x] with a, b ? C(R) and b ? a ? U(R); equivalent conditions for (x² ? 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.     

A Note on Lie Automorphisms,Subrings, and Lie Ideals of Prime Rings

ABSTRACT

Let R be a prime ring of characteristic not 2 or 3 and L a noncentral Lie ideal of R. Suppose that σ is a Lie automorphism on L such that σ² ? 1 is noncentral on L, where 1 is the identity map, then the subring of R generated by the set {[x ^σ, x] | x ∈ L} contains a nonzero ideal of R.     

X-Inner Automorphisms,Automorphisms, and Isomorphisms of Skew Polynomial Rings

Abstract

In this paper, we examine the X-inner automorphisms, automorphisms, and isomorphisms of skew polynomial rings of the form K[x][y;δ], where K is a field of characteristic 0 and δ is a derivation of K [x] such that x ^δ is a polynomial of degree ≥ 1. We 1.  determine the group of X-inner automorphisms of K [x][ y;δ],

2.  analyze the structure of the group of automorphisms of K [x][ y;δ], and

3.  examine the isomorphism classes of skew polynomial rings of the form K [x][ y;δ].

We also provide several examples which indicate the importance of the base field in computing the X-inner automorphisms, automorphisms, and isomorphism classes of K [x][ y;δ].     

An Example of Bergman's and the Extension Problem for Clean Rings

An example of Bergman is used to show that the extension of a clean ring by another clean ring need not be clean. That is, there exists a ring R and an ideal I of R such that both R/I and I are clean and idempotents lift modulo I, but R is not clean.     

On Lie Automorphisms,Additive Subgroups,and Lie Ideals of Prime Rings

Abstract

Let R be a prime ring of characteristic not equal to 2 or 3 and let L be a noncentral Lie ideal of R. Suppose that σ is a Lie automorphism on L such that σ⁴ is not the identity map. Then the additive subgroup generated by the set {[x ^σ, x] ∣ x ∈ L} contains a noncentral Lie ideal of R.