Tomasz Brzeziński, Christian Lomp
It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.
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A note on the differential smoothness of skew PBW extensions
本文研究了某族斜 Poincaré-Birkhoff-Witt 扩张的微分平滑性。
In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.
In this paper, we introduce the concept of $Σ$-semicommutative ring, for $Σ$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $Σ$-rigid, nil-reversible and rings satisfying the $Σ$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $Σ$-semicommutative rings. We prove that if a ring $R$ is $Σ$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-…
By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull dimension, prime length and global dimension in well-behaved cases, obtaining lower bounds to complement the upper bounds obtained by Venjakob and Wang.
A skew Laurent polynomial ring R[x^{\pm 1};α] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson …
We characterize skew polynomial rings and skew power series rings that are reduced and right or left Archimedean.
For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;τ_1,δ_1]...b[x_n;τ_n,δ_n] agrees with the PI degree of R[x_1;τ_1]...b[x_n;τ_n] when each (τ_i,δ_i) satisfies a q_i-skew relation for q_i \in k^{\times} and extends to a higher q_i-skew τ_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras.
We establish necessary or sufficient conditions to guarantee that skew Poincaré-Birkhoff-Witt extensions are NI or NJ rings. Our results extend those corresponding for skew polynomial rings and establish similar properties for other families of noncommutative rings such as universal enveloping algebras and examples of differential operators.
In this article, we work with unital twisted partial action of $\mathbb{Z}$ on an unital ring $R$ and we introduce the twisted partial skew power series rings and twisted partial skew Laurent series rings. We study primality, semi-primality and prime ideals in these rings. We completely describe the prime radical in partial skew Laurent series rings. Moreover, we study the Goldie property in partial skew power series rings and partial skew Laurent series rings and we describe conditions of the semiprimality of twisted partial skew power series rings.
Skew braces, near-rings, skew rings, dirings
本文引入左斜环和左弱环概念，通过将近环的单乘法替换为两个操作，研究Yang-Baxter方程的集合解。
For a ring endomirphism, we introduce the central skew Armendariz rings, which are a generalization of skew Armendariz rings and central Armendariz rings, and investigate their properties.
We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.
A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates a noetherian subring of R . F-noetherian rings have many interesting linear algebra properties which we refer to as the full strong rank condition, fully stably finite, and more generally the basic condition. We also study some basic ring-theoretic properties of F-noetherian rings such as localizations of F-noetherian rings. The F-noetherian property is pres…
Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study skew polynomial rings and skew power series rings over idempotent reflexive rings and abelian rings. Also, we introduce the concept of right (resp., left) $σ$-idempotent reflexive rings which generalizes right (resp., left) idempotent reflexive rings and $σ$-abelian rings. Certain results are obtained as corollaries from our results.
Given a group G acting on a ring R, one can construct the skew group ring R*G. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain more efficient proofs of the simplicity of certain rings constructed by H. Kambara and J. Trlifaj.
Let $α$ be an endomorphism of a ring $R$. We introduce the notion of weak $α$-skew McCoy rings which are a generalization of the $α$-skew McCoy rings and the weak McCo rings. Some properties of this generalization are established, and connections of properties of a weak $α$-skew McCoy ring $R$ with $n\times n$ upper triangular $T_n(R)$ are investigated. We study relationship between the weak skew McCoy property of a ring $R$ and it's polynomial ring, $R[x]$. Among applications, we show that a number of interesting properties of a weak $α$-skew McCoy ring $R$ such as weak skew McCoy property in…
Skew-convex function rings and evaluation of skew rational functions
本文利用斜多项式乘积的赋值公式构造了一类环，并基于此提出对斜多项式商进行赋值的方法。
In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of $Σ$-skew CN rings and $Σ$-skew reflexive (RNP) rings, for $Σ$ a finite family of ring endomorphisms of a ring $R$. Under certain conditions of compatibility, we study the transfer of the $Σ$-skew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corre…
Smooth geometry of skew PBW extensions over commutative polynomial rings I
研究交换多项式环上斜PBW扩张的微分光滑性，针对一个和两个不定元情形。
Smooth geometry of graded skew Clifford algebras
该论文聚焦分次斜Clifford代数领域，针对其微分光滑性展开研究，属于代数与微分几何交叉方向的理论探索。
We study the classification problem of possibly degenerate hermitian and skew hermitian bilinear forms over local rings where 2 is a unit.
The concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials, with coefficients over division rings. Iterated skew polynomial rings are in general not suitable for this purpose. Instead, multivariate skew polynomial rings are constructed in this work as follows: First, free multivariate skew polynomial rings are defined, where multiplication is additive on degrees and restricts to concatenation for monomials. This allows to define the evaluation of any skew polynomial at any point by unique remainder division. Multivariate skew pol…
In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields a…
Algebraic aspects of general free skew extensions of rings
本文研究环的自由斜扩张（即自由多元斜多项式环），给出其多种刻画，并分别给出其为整环、可嵌入级数环及素环的充分条件。
A ring $R$ is called a left APP-ring if the left annihilator $l_{R}(Ra)$ is right $s$-unital as an ideal of $R$ for any $a\in R$. Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω:S\longrightarrow {\rm End}(R)$ a monoid homomorphism. The skew generalized power series ring $[[R^{S,\leq},ω]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev-Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring $[[R^{S,\leq},ω]]$. It is shown that if $(S,\l…
On the annihilator condition for skew PBW extensions
本文研究了斜 PBW 扩张的零化子条件，证明了斜多项式环的相关结果同样适用，并指出零化子行为由扩张的定义关系决定，推广了已有结论。
We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier \cite{OuyangBirkenmeier2012} in the setting of noncommutative rings having PBW bases. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, differential operators on noncommutative spaces, noncommutative algebraic geometry, and theore…
Let $A$ be a ring and $φ$ its automorphism. It is proved that skew Laurent series ring $A((x,φ))$ is a right serial ring if and only if $A$ is a right serial right Artinian ring.
Given a partial (resp. a global) action $α$ of a connected finite groupoid $G$ on a ring $A$, we determine necessary and sufficient conditions for the partial (resp. global) skew groupoid ring $A\star_α G$ to be a separable extension of $A$.
Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over $RH$, we show that the skew group rings $RG$ and $RH$ share some properties such as being $n$-Gorenstein, $n$-perfect, $n$-coherent, $(n,d)$, Ding-Chen or IF-rings.
We study important invariants and properties of the Veronese subalgebras of $q$-skew polynomial rings, including their discriminant, center and automorphism group, as well as cancellation property and the Tits alternative.
As originally defined by Ashraf and Mozumder, multiplicative (generalized)-skew derivations must satisfy two identities. In this short note we show that, as a consequence of the simultaneous satisfaction of both identities, a multiplicative (generalized)-skew derivation of a prime ring is either a multiplicative (generalized) derivation (i.e., not skew), or a generalized skew derivation (i.e., additive). Therefore only one of the identities should be taken in the definition of multiplicative (generalized)-skew derivations in order to get a new class of derivations in prime rings.
We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
In this article, we show that for a partial skew group ring R*G, where R is a commutative ring, each non-zero ideal of R*G intersects R non-trivially if and only if R is a maximal commutative subring of R*G. As a consequence, we obtain necessary and sufficient conditions for simplicity; the partial skew group ring R*G is simple if and only if R is a G-simple and maximal commutative subring of R*G. We thereby generalize our previous results for skew group rings, to partial skew group rings which are not necessarily unital.
A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.
Let $α=(A_g,α_g)_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A=\bigoplus_{z\in G_0}A_z$ and $B=A\star_αG$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $λ$, we characterize when $λ$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general…
In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We 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 we 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 this elements and the principal coefficients are invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the ring of coefficients. Of course, this is valid for quantum polyno…
Let $X$ be a finite partially ordered set, $R$ an associative unital ring and $σ$ an endomorphism of $R$. We describe some properties of the skew incidence ring $I(X,R,σ)$ such as invertible elements, idempotents, the Jacobson radical and the center. Moreover, if the skew incidence rings $I(X,R,σ)$ and $I(Y,S,τ)$ are isomorphic and the only idempotents of $R,S$ are the trivial ones, we show that the partially ordered sets $X$ and $Y$ are isomorphic.
For an endomorphism s of R with s^{t}=1 we prove that the truncated polynomial ring (algebra) R[w,s]/(w^{t}) embeds into M_{t}(R[z]/(z^{t})). For an involution we exhibit an embedding of R into M_{2,1}^{s}(R), where M_{2,1}^{s}(R) is the algebra of the so called (s,2,1) supermatrices.
Let $X$ be a set of noncommuting variables of cardinality $card(X)\geqslant 2$, and ${\mathscr G}=\{σ_x\}_{x\in X}$, ${\mathscr D}=\{δ_x\}_{x\in X}$ be families of automorphisms and skew derivations of the ring $R$. It is proved that if the ring $R$ is semiprime Goldie, then the free skew extension $R[X;{\mathscr G},{\mathscr D}]$ is semiprimitive.
We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory, of the Cuntz-Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.
Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]…
Let A be a semprime, right noetherian ring equipped with an automorphism alpha, and let B := A[[y; alpha]] 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.
The strong b-lattice of quasi skew-rings
本文研究拟完全逆半环与强加法拟完全逆半环的关系，刻画后者构成拟斜环强 b-格的充要条件。
In this paper, we prove that if $R$ is an Archimedean reduced ring and satisfy ACC on annihilators, then $R[[x]]$ is also an Archimedean reduced ring. More generally we prove that if $R$ is a right Archimedean ring satisfying the \emph{ACC} on annihilators and $α$ is a rigid automorphism of $R$, then the skew power series ring $R[[x;α]]$ is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result.
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this short paper we study the d-Hermite condition about stably free modules for skew PBW extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and close related with these questions, …
Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study the transfert of the symmetry ($σ$-symmetry) and reversibility ($σ$-reversibility) from $R$ to its skew power series ring $R[[x;σ]]$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;σ]]$ in case $R$ is right $σ$-reversible. As a consequence we obtain a generalization of \cite{hong/2000}.
We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let $G$ be a finite group of graded algebra automorphisms of a skew polynomial ring $A:=k_{p_{ij}}[x_1,...,x_n]$. Then the fixed subring $A^G$ has finite global dimension if and only if $G$ is generated by quasi-reflections. In this case the fixed subring $A^G$ is isomorphic a skew polynomial ring with possibly different $p_{ij}$'s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.
The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
本文为非结合斜幂级数环和斜洛朗级数环建立希尔伯特基定理，通过引入拟结合性质给出Noetherian条件，并指出一般非结合情形下定理不成立。
Additivity of multiplicative (generalized) skew semi-derivations on rings
本文引入推广斜导与半导的新“斜半导”类，并研究此类乘法导子在环上可加的条件。
Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω: S\rightarrow End(R)$ a monoid homomorphism. In this paper we study the ascending chain conditions on principal left (resp. right) ideals of the skew generalized power series ring $R[[S,ω]]$. Among other results, it is shown that $R[[S,ω]]$ is a right archimedean reduced ring if $S$ is an Artinian strictly totally ordered monoid, $R$ is a right archimedean and $S$-rigid ring which satisfies the ACC on annihilators and $ω_s$ preserves nonunits of $R$ for each $s\in S$. As a consequence we deduce that the power series rings, Laurent …
We study skew inverse power series extensions R[[y^{-1};tau,delta]], where R is a noetherian ring equipped with an automorphism tau and a tau-derivation delta. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension, are all equal to 2n.
This paper is a natural continuation of the study of skew power series rings A initiated in [P. Schneider and O. Venjakob, On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2005), 349 - 367.]. We construct skew Laurent series rings B and show the existence of some canonical Ore sets S for the skew power series rings A such that a certain completion of the localisation A_S is isomorphic to B. This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings.
On Weakly Separable Polynomials in Skew Polynomial Rings
本文研究了斜多项式环中弱可分离多项式的特征化及其与可分离性的关系。
The Eliahou-Kervaire resolution over a skew polynomial ring
本文将Eliahou和Kervaire在多项式环中为稳定理想构造的最小分解推广到斜多项式环，证明其同调性质仍成立，并给出推广的乘积结构。
In this paper we proved that if $R$ is right $ω$-Euclidean domain, then skew Laurent formal series ring is right $ω$-Euclidean domain. We also showed that if $R$ is a right $ω$-Euclidean domain with multiplicative norm, then skew Laurent formal series ring is a right principal ideal domain. In addition, we proved that if $R$ is a noncommutative $ω$-Euclidean domain with a multiplicative norm, then $R$ and skew Laurent formal series ring is a ring with elementary reduction of matrices.
We observe that the Dieudonné determinant induces a non-negative degree function on the ring of matrices over a skew polynomial ring. We then apply this degree function to two examples. In the first one, we find an expression for the rank of the kernel of an endomorphism of the algebraic group $\mathbb{G}_a^n$ over a field of characteristic $p>0$. In the second, we calculate the dimension of the solution space of linear matrix differential equations.
Skew Generalized Power Series Rings With the McCoy Property
本文研究了斜广义幂级数环上的 $(S,ω)$-McCoy 条件，将标准 McCoy 条件从多项式推广至斜广义幂级数，并推广了相关现有结果。
Let $R$ be a ring satisfying a polynomial identity and let $D$ be a derivation of $R$. We consider the Jacobson radical of the skew polynomial ring $R[x;D]$ with coefficients in $R$ and with respect to $D$, and show that $J(R[x;D])\cap R$ is a nil $D$-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when $R$ is commutative.
We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.
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