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.
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.
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] i…
Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. We obtain necessary and sufficient conditions for when this idealiser subring is left and right noetherian. We also give an example of these conditions in practice which translates to an interesting number theoretic problem.
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.
We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced …
For a field $F$ of characteristic not 2 and a directed row-finite graph $Γ$ let $L(Γ)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(Γ),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.
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…
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.
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.
Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring RP with respect to a prime ideal P of R and we prove a criterion about the D- simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author. Second we construct …
Let R0 be a commutative associative ring (not necessarily unital), G a group and alpha a partial action by ideals that contain local units. We show that R0 is maximal commutative in the partial skew group ring R0*G if and only if R0 has the ideal intersection property in R0*G. From this we derive a criterion for simplicity of R0*G in terms of maximal commutativity and $G-$simplicity of R0 and apply this to two examples, namely to partial actions by clopen subsets of a compact set and to give a new proof of the simplicity criterion for Leavitt path algebras. A new proof of the Cuntz-Krieger uni…
Given a partial action $π$ of an inverse semigroup $S$ on a ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_πS$. Our main result asserts that, when $\mathcal{A}$ is commutative, the ring $\mathcal{A} \rtimes_πS$ is simple if, and only if, $\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_πS$ and $\mathcal{A}$ is $S$-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore,…
Let $\A$ be a ring with local units, $\E$ a set of local units for $\A$, $\G$ an abelian group and $α$ a partial action of $\G$ by ideals of $\A$ that contain local units and such that the partial skew group ring $\A\star_α \G$ is associative. We show that $\A\star_α \G$ is simple if and only if $\A$ is $\G$-simple and the center of the corner $eδ_0 (\A\star_α \G) e δ_0$ is a field for all $e\in \E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and par…
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.
Automorphic Transcendence Degree, Growth, and Skew Normalization
Faculty of Advanced Education, Ho Chi Minh City University of Technology and Engineering（胡志明市工程技术大学高级教育学院）；Department of Computer Science, University of Bath（巴斯大学计算机系）；Analytical and Algebraic Methods in Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University（顿铎成大学数学与统计学院优化分析代数方法研究组）
Given a group G, a (unital) ring A and a group homomorphism $σ: G \to \Aut(A)$, one can construct the skew group ring $A \rtimes_σ G$. We show that a skew group ring $A \rtimes_σ G$, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. If G is abelian and A is commutative, then $A \rtimes_σ G$ is shown to be simple if and only if σis injective and A is G-simple. As an application we show that a transformation group (X,G), where X is a compact Hausdorff space and G is abelian, is minimal and faithful if and only if its associated skew group algebra $C(X) \rti…
A skew polynomial ring $R=K[x;σ,δ]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. There is also a structure of conjugacy classes on the roots. In $R=F_{q^m}[x,σ]$, this leads to matroids of right independent and left independent sets. These matroids are isomorphic via the extension of the map $ϕ:[1]\to[1]$ defined by $ϕ(a)=…
A ring $R$ is said to be i-reversible if for every $a,b$ $\in$ $R$, $ab$ is a non-zero idempotent implies $ba$ is an idempotent. It is known that the rings $M_n(R)$ and $T_n(R)$ (the ring of all upper triangular matrices over $R$) are not i-reversible for $n \geq 3$. In this article, we provide a non-trivial i-reversible subring of $M_n(R)$ when $n \geq 3$ and $R$ has only trivial idempotents. We further provide a maximal i-reversible subring of $T_n(R)$ for each $n\geq 3$, if $R$ is a field. We then give conditions for i-reversibility of Trivial, Dorroh and Nagata extensions. Finally, we give…
We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity o…
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.
We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element $z$. In the case where this element exists we give simplicity criteria for the rings obtained by inverting $z$ and the rings obtained by factoring out the ideal generated by $z$. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.
We obtain sufficient criteria for simplicity of systems, that is, rings $R$ that are equipped with a family of additive subgroups $R_s$, for $s \in S$, where $S$ is a semigroup, satisfying $R = \sum_{s \in S} R_s$ and $R_s R_t \subseteq R_{st}$, for $s,t \in S$. These criteria are specialized to obtain sufficient criteria for simplicity of, what we call, s-unital epsilon-strong systems, that is systems where $S$ is an inverse semigroup, $R$ is coherent, in the sense that for all $s,t \in S$ with $s \leq t$, the inclusion $R_s \subseteq R_t$ holds, and for each $s \in S$, the $R_s R_{s^*}$-$R_{…
In this article, we consider a twisted partial action $α$ of a group $G$ on a ring $R$ and it is associated partial crossed product $R*_α^wG$. We study necessary and sufficient conditions for the commutativity and simplicity of $R*_α^wG$. Let $R=C(X)$ the algebra of continuous functions of a topological space $X$ on the complex numbers and $C(X)*_αG$ the partial skew group ring, where $α$ is a partial action of a topological group $G$ on $C(X)$. We study some topological properties to obtain results on the algebra $C(X)$. Also, we study the simplicity of $C(X)*_αG$ using topological properties…
Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusionof R into S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce …
In this article we introduce the notion of a controlled group graded ring. Let $G$ be a group, with identity element $e$, and let $R=\oplus_{g\in G} R_g$ be a unital $G$-graded ring. We say that $R$ is $G$-controlled if there is a one-to-one correspondence between subsets of the group $G$ and (mutually non-isomorphic) $R_e$-bimodules in $R$, given by $G \supseteq H \mapsto \oplus_{h\in H} R_h$. For strongly $G$-graded rings, the property of being $G$-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general $G$-graded ring to be $G$-controll…
Given a partial action $α$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_α G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_αG.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_αG.$ We show that every ideal of $R \rtimes_αG$ is graded if, and only if, $α$ has the residual intersection property. Furthermore, if $α$ is induced by a topological …
We solve the problem of extension of characters of commutative subalgebras in associative (noncommutative) algebras for a class of subrings (Galois orders) in skew group rings. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras, Yangians and finite $W$-algebras. In particular, we develop a theory of Gelfand-Tsetlin modules for $\gl_n$. B…
For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying AI \subseteq IA. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-n…
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the commutant of the neutral component subring $R_e$ in $R$ and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant ideals). We also show that if $R_e$ is commutative (not necessarily maximal com…
A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large c…
Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and α\in \Mat_n(KG). Then αinduces a bounded linear map α: l^2(G)^n \to ł^2(G)^n,…
