arXiv subjects
Publications and source records attributed to Johan Öinert.
In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;δ]$, under the hypothesis that $R$ is $s$-unital and $\ker(δ)$ contains a nonzero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.
Let $N$ and $H$ be groups, and let $G$ be an extension of $H$ by $N$. In this article we describe the structure of the complex group ring of $G$ in terms of data associated with $N$ and $H$. In particular, we present conditions on the building blocks $N$ and $H$ guaranteeing that $G$ satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison-Kaplansky conjecture holds for the group C*-algebra of $G$.
In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime $s$-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over $s$-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime $s$-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.
Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.
Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this article, we show that $L_R(E)$ is always nearly epsilon-strongly $G$-graded. We also show that if $E$ is finite, then $L_R(E)$ is epsilon-strongly $G$-graded. We present a new proof of Hazrat's characterization of strongly $\mathbb{Z}$-graded Leavitt path algebras, when $E$ is finite. Moreover, if $E$ is row-finite and has no source, then we show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink. We also use a result concerning Frobenius epsilon-strongly $G$-graded rings, where $G$ is finite, to obtain criteria which ensure that $L_R(E)$ is Frobenius over its identity component.
Given a non-associative unital ring $R$, a monoid $G$ and a set $π$ of additive maps $R \rightarrow R$, we introduce the Ore monoid ring $R[π; G]$, and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called $D$-structures $π$. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.
We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group graded situation to the groupoid graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.
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, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra $A_R(\mathcal{G})$ associated with a Hausdorff and ample groupoid $\mathcal{G}$.
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988.
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 $G$-controlled. We also give a characterization of strongly $G$-graded rings which are $G$-controlled. As an application of our main results we give a description of all intermediate subrings $T$ with $R_e \subseteq T \subseteq R$ of a $G$-controlled strongly $G$-graded ring $R$. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.
In this article we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide non-trivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.
Let $α= \{ α_g : R_{g^{-1}} \rightarrow R_g \}_{g \in \textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \star_α G$ be the associated partial skew groupoid ring. We show that if $α$ is global and unital, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely many $g \in \textrm{mor}(G)$. We use this result to prove that if $α$ is unital and $R$ is alternative, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely many $g \in \textrm{mor}(G)$. Both of these results apply to partial skew group rings, and in particular they generalize a result by J. K. Park for classical skew group rings, i.e. the case when $R$ is unital and associative, and $G$ is a group which acts globally on $R$. Moreover, we provide two applications of our main result. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (non-associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we use globalization to analyse noetherianity and artinianity of partial skew groupoid rings as well as establishing two Maschke-type results, thereby generalizing results by Ferrero and Lazzarin from the group graded case to the groupoid situation.
We introduce non-associative Ore extensions, $S = R[X ; σ, δ]$, for any non-associative unital ring $R$ and any additive maps $σ,δ: R \rightarrow R$ satisfying $σ(1)=1$ and $δ(1)=0$. In the special case when $δ$ is either left or right $R_δ$-linear, where $R_δ = \ker(δ)$, and $R$ is $δ$-simple, i.e. $\{ 0 \}$ and $R$ are the only $δ$-invariant ideals of $R$, we determine the ideal structure of the non-associative differential polynomial ring $D = R[X ; \mathrm{id}_R , δ]$. Namely, in that case, we show that all ideals of $D$ are generated by monic polynomials in the center $Z(D)$ of $D$. We also show that $Z(D) = R_δ[p]$ for a monic $p \in R_δ[X]$, unique up to addition of elements from $Z(R)_δ$. Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of $D$ to show that $D$ is simple if and only if $R$ is $δ$-simple and $Z(D)$ equals the field $R_δ \cap Z(R)$. This provides us with a non-associative generalization of a result by Öinert, Richter, and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of $D$ in the cases when the characteristic of the field $R_δ \cap Z(R)$ is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.
We extend the classicial notion of an outer action $α$ of a group $G$ on a unital ring $A$ to the case when $α$ is a partial action on ideals, all of which have local units. We show that if $α$ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A \star_αG$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.
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 of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of D. Gonçalves'. We also point out how E. Jespers' result immediately implies a generalization of a simplicity result, recently obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by twisted partial actions.
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-negative integers satisfying the following two conditions (d1) if a \in A, then d(a)=0 if and only if a=0; (d2) there is a subset X of B generating B as a ring such that for each non-zero ideal I of A and each non-zero a \in I there is a non-zero a' \in I with d(a') \leq d(a) and d(a'b - ba') < d(a) for all b \in X. We show that if the centralizer C of B in A is an A-simple ring, every intersection of C with an ideal of A is A-invariant, ACA=A and there is a degree map for A/B, then A is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings.
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.