Goldie Conditions for Ore Extensions over Semiprime Rings

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension. Presented by S. Montgomery Mathematics Subject Classification (2000) 16S90. Jerzy Matczuk: Supported by the Flemish–Polish bilateral agreement BIL 01/31.     

Ore Extensions of Quasi-Baer Rings

We first study the quasi-Baerness of R[x; σ, δ] over a quasi-Baer ring R when σ is an automorphism of R, obtaining an affirmative result. We next show that if R is a right principally quasi-Baer ring and σ is an automorphism of R with σ(e) = e for any left semicentral idempotent e ∈ R, then R[x; σ, δ] is right principally quasi-Baer. As a corollary, we have that R[x; δ] over a right principally quasi-Baer ring R is right principally quasi-Baer. Finally, we give conditions under which the quasi-Baernesses (right principal quasi-Baernesses) of R and R[x; σ, δ] are equivalent.     

On Induced Modules Over Ore Extensions

Abstract

Let R be a ring and S = R[x;σ, δ] its Ore extension. For an R-module M _R we investigate the uniform dimension and associated primes of the induced S-module M ? _R  S.     

Hopf代数的二重Ore扩张

本文的目的 是定义Hopf二重Ore扩张,讨论这种扩张的基本性质并研究Hopf代数的分次与Hopf二重Ore扩张之间的关系.作者还研究了连通分次Hopf代数的结构及其Hopf二重Ore扩张的同调性质.     

Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U _s (R)[X; D], where U _s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U _s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U _s (R) such that δ(α) = 0 for all δ ? D. These extend the known results for free algebras.     

Hochschild and Cyclic Homology of Ore Extensions and Some Examples of Quantum Algebras

For every Ore extension we construct a chain complex giving its Hochschild homology. As an application we compute the Hochschild and cyclic homology of an arbitrary multiparametric affine space and the Hochschild homology of the algebra of differential operators over this space, in the generic case.     

Ore Extensions of Ore Domains

Let R be a right Ore domain and φ a derivation or an automorphism of R. We determine the right Martindale quotient ring of the Ore extension R[t; φ] (Theorem 1.1). As an attempt to generalize both the Weyl algebra and the quantum plane, we apply this to rings R such that k[x] ? R ? k(x), where k is a field and x is a commuting variable. The Martindale Quotient quotient ring of R[t; φ] and its automorphisms are computed. In this way, we obtain a family of non-isomorphic infinite dimensional simple domains with all their automorphisms explicitly described.     

Fibers in Ore Extensions

Let R be a finitely generated commutative algebra over an algebraically closed field k and let A=R[t;,] be the Ore extension with respect to an automorphism and a -derivation . We view A as the coordinate ring of an affine noncommutative space X. The inclusion RA gives an affine map : XSpecR, and X is a noncommutative analogue of A ¹×SpecR. We define the fiber X _p of over a closed point pSpecR as a certain full subcategory ModX _p of ModA. The category ModX _p has the following structure. If p has infinite -orbit, then ModX _p is equivalent to the category of graded modules over the polynomial ring k[x] with degx=1. If p is not fixed by , but has finite -orbit, say of size n, then ModX _p is equivalent to the representations of the quiver à _n–1 with the arrows all going in the same direction. If p is fixed by , then ModX _p is equivalent to either Modk or Modk[x]. It is also shown that X is the disjoint union of the fibers X _p in a certain sense.     

拟三角Hopf代数的Ore -扩张

该文主要考虑了拟三角Hopf代数的某种Ore -扩张问题. 对拟三角Hopf代数的Ore -扩张何时保持相同的拟三角结构给出了充分必要条件. 最后作为应用, 文章讨论了Sweedler Hopf代数和Lusztig小量子群的Ore -扩张结构.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

Iterated conditional expectations

Consider the probability space ([0,1),B,λ), where B is the Borel σ-algebra on [0,1) and λ the Lebesgue measure. Let f=₁[0,1/2) and g=₁[1/2,1). Then for any ε>0 there exists a finite sequence of sub-σ-algebras Gj⊂B(j=1,…,N), such that putting f₀=f and fj=E(fj−1|Gj), j=1,…,N, we have ‖fN−g_‖∞<ε; here E(⋅|Gj) denotes the operator of conditional expectation given σ-algebra Gj. This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.     

On the structure of semi-invariant polynomials in Ore extensions

Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X;S,D] is ν if charR=0, and is p^kν for some k0 if charR=p2, where ν is the least integer ν1 such that S^νDS^−ν−D is X-inner. A similar result holds for cv-polynomials. These are done by introducing the new notion of k-basic polynomials for each integer k0, which enable us to analyze semi-invariant polynomials inductively.     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

The Generalized Center Galois Extensions of Rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and I_i = {c - g_i(c) | c C} for each g_i G. Then, B is called a center Galois extension with Galois group G if BI_i = B for each g_i 1 in G, and a weak center Galois extension with group G if BI_i = Be_i for some nonzero idempotent e_i in C for each g_i 1 in G. When e_i is a minimal element in the Boolean algebra generated by {e_i | g_i G} Be_i is a center Galois extension with Galois group H_i for some subgroup H_i of G. Moreover, the central Galois algebra B(1 – e_i) is characterized when B is a Galois algebra with Galois group G.AMS Subject Classification (1991): 16S35 16W20Supported by a Caterpillar Fellowship, Bradley University, Peoria, Illinois, USA. We would like to thank Caterpillar Inc. for their support.     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.     

$2$-Primal环的Ore扩张中的极小素理想和单位元

Let R be an(α,δ)-compatible ring.It is proved that R is a 2-primal ring if and only if for every minimal prime ideal P in R[x;α,δ] there exists a minimal prime ideal P in R such that P = P [x;α,δ],and that f(x) ∈ R[x;α,δ] is a unit if and only if its constant term is a unit and other coefficients are nilpotent.     

Azumaya Extensions and Galois Correspondence

For H a finite-dimensional Hopf algebra over a field k, we study H*-Galois Azumaya extensions A, i.e., A is an H-module algebra which is H*-Galois with A/A^H separable and A^H Azumaya. We prove that there is a Galois correspondence between a set of separable subalgebras of A and a set of separable subalgebras of C_A(A^H), thus generalizing the work of Alfaro and Szeto for H a group algebra. We also study Galois bases and Hirata systems.1991 Mathematics Subject Classification: 16W30, 16H05     

PC—环与几乎优越扩张

本文我们讨论了这种环R的结构,R是凝聚环且R作为R－模是P－内射,我们称此环为PC－环,并证明了在几乎优越扩张下的不变性。