Given a partial action π of an inverse semigroup S on a ring A {\mathcal{A}}, one may construct its associated skew inverse semigroup ring A π S {\mathcal{A}\rtimes-{\pi}S}. Our main result asserts that, when A {\mathcal{A}} is commutative, the ring A π S {\mathcal{A}\rtimes-{\pi}S} is simple if, and only if, A {\mathcal{A}} is a maximal commutative subring of A π S {\mathcal{A}\rtimes-{\pi}S} and A {\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 (g) {A-{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid g {\mathcal{G}}. © 2018 Walter de Gruyter GmbH, Berlin/Boston.
In this article we give a characterization of left (right) quasi-duo differential polynomial rings. 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.
Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R, denoted by LR(E). For an arbitrary group G, LR(E) can be viewed as a G-graded ring. In this paper, we show that LR(E) is always nearly epsilon-strongly G-graded. We also show that if E is finite, then LR(E) is epsilon-strongly G-graded. We present a new proof of Hazrat's characterization of strongly g-graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that LR(E) is strongly-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 LR(E) is Frobenius over its identity component. © 2020 World Scientific Publishing Company.
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 from the associative case to the non-associative situation. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply this 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.
Let α={α_g : R_{g^{−1}}→R_g}_{g∈mor(G)} be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=R⋆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∈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∈mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem 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. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily 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 relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the collection of strongly graded rings and the family of unital partial crossed products. We determine when epsilon-strongly graded rings are separable over their principal components. 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 family 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 family 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 Simon concerning when graded rings can be presented as partial crossed products.
We introduce non-associative Ore extensions, S = R[X; sigma, delta], for any non-ssociative unital ring R and any additive maps sigma, delta : R -> R satisfying sigma(1) = 1 and delta(1) = 0. In the special case when delta is either left or right R-delta-linear, where R-delta = ker(delta), and R is delta-simple, i.e. {0} and R are the only delta-invariant ideals of R, we determine the ideal structure of the non-associative differential polynomial ring D = R[X; id(R),delta]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z(D) = R-delta[p] for a monic p is an element of R-delta [X], unique up to addition of elements from Z(R)(delta) . 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 delta-simple and Z(D) equals the field R-delta boolean AND Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silve-strov. 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-delta boolean AND 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.
Given a non-associative unital ring R, a monoid G and a set π of additive maps R→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. © 2019 Elsevier Inc.
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 = aS center dot (gaG) 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) -sub-bimodules of R, given by G aSc Ha dagger broken vertical bar aS center dot (haH) 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) aS dagger T aS dagger 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.