On the separability of the partial skew groupoid ring

@article{Bagio2017OnTS,
  title={On the separability of the partial skew groupoid ring},
  author={Dirceu Bagio and H. Pinedo},
  journal={S{\~a}o Paulo Journal of Mathematical Sciences},
  year={2017},
  volume={11},
  pages={370-384}
}
Given a partial (resp. a global) action $$\alpha $$α of a connected finite groupoid $$\mathscr {G}$$G on a ring $$\mathcal {A},$$A, we determine necessary and sufficient conditions for the partial (resp. the global) skew groupoid ring $$\mathcal {A}\star _\alpha \mathscr {G}$$A⋆αG to be a separable extension of $$\mathcal {A}$$A. 
On partial skew groupoids rings
Given a partial action $\alpha$ of a connected groupoid $\mathcal{G}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A\star_{\alpha}\mathcal{G}$ canExpand
Artinian and noetherian partial skew groupoid rings
Let $\alpha = \{ \alpha_g : R_{g^{-1}} \rightarrow R_g \}_{g \in \textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \star_{\alpha} G$ be theExpand
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 aExpand
On partial skew groupoid rings
Given a partial action α of a connected groupoid 𝒢 on an associative ring A we investigate under what conditions the partial skew groupoid ring A ⋆α𝒢 can be realized as a partial skew group ring.Expand
Isomorphism Theorems for Groupoids and Some Applications
TLDR
Inspired by the Ehresmann-Schein-Nambooripad theorem, a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups is improved. Expand
Recent developments around partial actions
We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups,Expand
Partial groupoid actions on R-categories: Globalization and the smash product
In this paper, we introduce the concept of partial groupoid actions on R-semicategories as well as we give criteria for existence of a globalization of it. This point of view is a generalization of...
Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras
Let α = (Ag, αg)g∈G be a group-type partial action of a connected groupoid G on a ring A = ⊕ z∈G0 Az and B := A ⋆α G the corresponding partial skew groupoid ring. In the first part of this paper weExpand
Inverse Semigroupoid Actions and Representations.
We show that there is a one-to-one correspondence between the partial actions of a groupoid $G$ on a set $X$ and the inverse semigroupoid actions of the Exel's inverse semigroupoid $S(G)$ on $X$.Expand
Object-unital groupoid graded rings, crossed products and separability
Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal)Expand
...
1
2
...

References

SHOWING 1-10 OF 15 REFERENCES
PARTIAL ACTIONS OF ORDERED GROUPOIDS ON RINGS
In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under whichExpand
The structure of Frobenius algebras and separable algebras
We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equationExpand
Crossed Products by Twisted Partial Actions: Separability, Semisimplicity, and Frobenius Properties
In this article, we are concerned with the crossed product S = R☆α G by a twisted partial action α of a group G on a ring R. We give necessary and sufficient (resp., sufficient) conditions for S toExpand
Associativity of crossed products by partial actions, enveloping actions and partial representations
Given a partial action a of a group G on an associative algebra A, we consider the crossed product A × α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on theExpand
Separable Groupoid Rings
We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring ofExpand
Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory
In this article, we introduce the notion of a partial action of a groupoid on a ring as well as we give a criteria for the existence of a globalization of it. We construct a Morita context associatedExpand
Skew group rings which are azumaya
If S is a ring with 1, and G is a finite group acting faithfully as automorphisms of S, then it is well known that the skew group ring S ∗ G is a separable extension of S if and only if there existsExpand
Partial actions and Galois theory
Abstract In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by S.U. Chase, D.K. HarrisonExpand
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 determineExpand
Crossed product algebras defined by separable extensions
Abstract We generalize the classical construction of crossed product algebras defined by finite Galois field extensions to finite separable field extensions. By studying properties of rings graded byExpand
...
1
2
...