| 1. |
- Cala, Juan, et al.
(författare)
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Graded modules over object-unital groupoid graded rings
- 2021
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Ingår i: Communications in Algebra. - : Taylor & Francis Group. - 0092-7872 .- 1532-4125. ; 50:2, s. 444-462
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we analyze the category(Formula presented) of unitary G-graded modules over object unital G -graded rings R, being G a groupoid. Here we consider the forgetful functor G - R- mod and determine many properties (Formula presented.) for which the following implications are valid for modules M in (Formula presented.) M is (Formula presented.) (Formula presented.) U(M) is (Formula presented.) U(M) is (Formula presented.) (Formula presented.) M is (Formula presented.) We treat the cases when (Formula presented.) is any of the properties: direct summand, projective, injective, free and semisimple. Moreover, graded versions of results concerning classical module theory are established, as well as some structural properties (Formula presented.).
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| 2. |
- Cala, Juan, et al.
(författare)
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Object-unital groupoid graded rings, crossed products and separability
- 2021
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Ingår i: Communications in Algebra. - : Informa UK Limited. - 0092-7872 .- 1532-4125. ; 44:4, s. 1676-1696
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Tidskriftsartikel (refereegranskat)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) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation. © 2020 The Author(s). Published with license by Taylor and Francis Group, LLC.
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| 3. |
- Lundström, Patrik, et al.
(författare)
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Very good gradings on matrix rings are epsilon-strong
- 2024
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Ingår i: Linear and multilinear algebra. - : Taylor & Francis. - 0308-1087 .- 1563-5139.
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Tidskriftsartikel (refereegranskat)abstract
- We investigate properties of group gradings on matrix rings (Formula presented.), where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on (Formula presented.) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that (Formula presented.) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when (Formula presented.) is an epsilon-crossed product. Our results are illustrated by several examples. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
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| 4. |
- Nystedt, Patrik, 1971-, et al.
(författare)
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Artinian and noetherian partial skew groupoid rings
- 2018
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Ingår i: Journal of Algebra. - : Academic Press. - 0021-8693 .- 1090-266X. ; 503, s. 433-452
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Nystedt, Patrik, 1971-, et al.
(författare)
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Epsilon-strongly graded rings, separability and semisimplicity
- 2018
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 514, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- 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 NÇŽstÇŽsescu, 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 appears 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. © 2018 Elsevier Inc.
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| 6. |
- Nystedt, Patrik, 1971-, et al.
(författare)
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EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY
- 2020
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Ingår i: Glasgow Mathematical Journal. - : CAMBRIDGE UNIV PRESS. - 0017-0895 .- 1469-509X. ; 62:1, s. 233-259
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Tidskriftsartikel (refereegranskat)abstract
- 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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