Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

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.     

On Semiprime Goldie Modules

In this article, we investigate some properties of right core inverses. Particularly, new characterizations and expressions for right core inverses are given, using projections and {1, 3}-inverses. Also, we introduced and investigated a new generalized right core inverse which is called right pseudo core inverse. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses, and EP elements.     

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime.     

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.     

Partial Crossed Products and Goldie Rings

In this paper we consider the transfer of the property of being a left Goldie ring between a ring A and its partial crossed product A*_α G by a twisted partial action α of a group G on A.     

Ore Extensions over Total Valuation Rings

It is shown that any Ore extension R = V[x;σ,δ] over a total valuation ring V is always v-Bezout which is a generalization of commutative GCD domains. By using this result, a necessary and sufficient condition are given for R to be fully left bounded.     

半质环的交换性条件

本文对满足可变恒等式的半质环在某种有界条件下给出了一个判断环Ｒ交换性的简便准则，使文献［２－２７］中所有相应结果均成为其直接推论．此外，对不限有界的情况，也得到较为广泛的结论．     

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.     

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.     

Automorphisms for Skew PBW Extensions and Skew Quantum Polynomial Rings

In this work, I study the automorphisms of skew PBW extensions and skew quantum polynomials. I use Artamonov's works as reference for getting the principal results about automorphisms for generic skew PBW extensions and generic skew quantum polynomials. In general, if I have an endomorphism on a generic skew PBW extension and there are some x _i , x _j , x _u such that the endomorphism is not zero on these elements and the principal coefficients are invertible, then endomorphisms act over x _i as a _i x _i for some a _i in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with r = 0, as such Artamonov shows in his work. We use this result for giving some more general results for skew PBW extensions, using filtred-graded techniques. Finally, I use localization to characterize some class the endomorphisms and automorphisms for skew PBW extensions and skew quantum polynomials over Ore domains.     

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.     

Goldie Ranks of Skew Power Series Rings of Automorphic Type

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B: = A[[y; α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.     

半素环的几个交换性条件

一个半素环 R是交换环当且仅当 R满足下列条件之一 :( ) (xmy) n+xmy∈ Z(R) ,对任意的 x ,y∈ R。( ) (xmy) n- yxm∈ Z(R) ,对任意的 x,y∈ R。( ) (xmy) n+yxm∈ Z(R) ,对任意的 x,y∈ R。其中 m,n是固定的正整数且 n >1     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

On pure Goldie dimensions

In this paper, we examine the pure Goldie dimension and dual pure Goldie dimension in finitely accessible additive categories. In particular, we show that if A is an object in a finitely accessible additive category 𝒜 that has finite pure Goldie dimension n and finite dual pure Goldie dimension m, then End_𝒜(A) is semilocal and the dual Goldie dimension of End_𝒜(A) is less than or equal to n+m.