The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second subject consists of investigating the class of S-modules which are finitely generated as R-module. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.
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.
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.
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.
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…
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…
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…
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…
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.
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.
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.
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.
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.
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.
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
Mindanao State University-Iligan Institute of Technology（棉兰老州立大学伊利甘理工学院）；Blekinge Institute of Technology（布莱金厄理工学院）
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.
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.
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-…
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.
Different approaches to construct linear codes using skew polynomials can be unified by using the nonassociative algebras built from skew-polynomial rings by Petit.
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.
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 study the interplay between Steinberg algebras and partial skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorf…
In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg.
We study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,σ, δ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a \in D$ and $f \in D[x,σ,δ]$ satisfy $f(a) = a$, then $f^{\circ n}(a) = a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.
For a ring $R$ with an automorphism $σ$, an $n$-additive mapping $Δ:R\times R\times... \times R \rightarrow R$ is called a skew $n$-derivation with respect to $σ$ if it is always a $σ$-derivation of $R$ for each argument. Namely, it is always a $σ$-derivation of $R$ for the argument being left once $n-1$ arguments are fixed by $n-1$ elements in $R$. In this short note, starting from Brešar Theorems, we prove that a skew $n$-derivation ($n\geq 3$) on a semiprime ring $R$ must map into the center of $R$.
In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.
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.
In this paper we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew $PBW$ extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand-Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre's conjecture will be discussed for a particular class of skew $PBW$ extensions. The questions about the noetherianity and the Zariski cancellation property of Artin-Schelter regular algebras will be reformulated f…
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.
Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case $S$ is a subsemigroup of a group $G$ such that $G=S^{-1}S$, we obtain a $G$-graded ring $S^{\rm op} * A * S$ with the property that, for each $s\in S$, the $s$-component contains a left invertible element and the $s^{-1}$-component contains a right invertible element. In the most basic case, where $G$ is the additive grou…
