On partial Galois abelian extensions
In this article we construct the inverse semigroup of equivalence classes of partial Galois abelian extensions of a commutative ring R with same group G, called the Harrison partial inverse semigroup.
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arXiv subjects
Héctor Pinedo
Publications and source records attributed to Héctor Pinedo.
Partial actions and cyclic Kummer's theory
We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by A. Z. Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial $n$-kummerian extension is equivalent to either a radical or a $I$-radical extension, for some subgroup $I$ of the cyclic group $C_n$.
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Isomorphism Theorems for Groupoids and Some Applications
Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities, and study subgroupoids, wide subgroupoids and normal subgroupoids. We also present the isomorphism theorems for groupoids and as an applications we obtain the corresponding version of Zassenhaus Lemma and Jordan-Hölder Theorem for groupoids. Finally inspired by the Ehresmann-Schein-Nambooripad Theorem we improve a result of R. Exel concerning a one to one correspondence between partial actions of groups and actions of inverse semigroups.
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On partial skew groupoids rings
Given a partial action $α$ of a connected groupoid $\mathcal{G}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A\star_α\mathcal{G}$ can be realized as a partial skew group ring. In such a case applications concerning to the separability, semisimplicity and Frobenius property of the ring extension $A\subset A\star_α\mathcal{G}$ as well as to the artinianity of $A\star_α\mathcal{G}$ are given.
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Epsilon-strongly groupoid graded rings, the Picard inverse category and cohomology
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.
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Lifting partial actions: from groups to groupoids
In this paper, we are interested in the study of the existence of connections between partial groupoid actions and partial group actions. Precisely, we prove that there exists a datum connecting a partial action of a connected groupoid and a partial action of any of its isotropy groups. Furthermore, it will be proved that under a suitable condition the partial skew groupoid ring corresponding to a partial action by a connected groupoid is isomorphic to a specific partial skew group ring. We also present a Morita theory and a Galois theory related to these partial actions as well as considerations about the strictness of the corresponding Morita contexts. Semisimplicity, separability and Frobenius properties of the corresponding partial skew groupoid rings are also considered.
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Epsilon-strongly graded rings, separability and semisimplicity
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.
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Artinian and noetherian partial skew groupoid rings
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.
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