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  • 51.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.
    Öinert, Johan
    Lund University, Centre for Mathematical Sciences.
    Outer Partial Actions and Partial Skew Group Rings2015In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-4279, Vol. 67, no 5, p. 1144-1160Article in journal (Refereed)
    Abstract [en]

    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⋆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.

  • 52.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Tekniska Högskola, Karlskrona, Sweden.
    Simple graded rings, nonassociative crossed products and Cayley-Dickson doublings2020In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 19, no 12, article id 2050231Article in journal (Refereed)
    Abstract [en]

    e show that if a nonassociative 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 nonassociative setting. By applying this result to nonassociative crossed products, we obtain nonassociative 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. © 2020 World Scientific Publishing Company.

  • 53.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Öinert, Johan
    University of Copenhagen, Department of Mathematical Sciences.
    Simple skew category algebras associated with minimal partially defined dynamical systems2013In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 33, no 9, p. 4157-4171Article in journal (Refereed)
    Abstract [en]

    In this article, we continue our study of category dynamical systems, that is functors s from a category G to Topop, and their corresponding skew category algebras. Suppose that the spaces s(e), for e ∈ ob(G), are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) G is inverse connected, (iii) s is minimal and (iv) s is faithful. We also show that if G is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.

  • 54.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Inst Technol, Dept Math & Nat Sci, SE-37179 Karlskrona, Sweden.
    Pinedo, Hector
    Univ Ind Santander, Escuela Matemat, Carrera 27 Calle 9, Bucaramanga, Colombia (COL).
    Epsilon-Strongly Groupoid-Graded Rings, The Picard Inverse Category And Cohomology2020In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 62, no 1, p. 233-259Article in journal (Refereed)
    Abstract [en]

    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.

  • 55.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.
    Pinedo, Héctor
    Universidad Industrial de Santander, Escuela de Matemáticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia.
    Artinian and noetherian partial skew groupoid rings2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, p. 433-452Article in journal (Refereed)
    Abstract [en]

    Let alpha = {alpha(g) : Rg-1 -> R-g}(g is an element of mor(G)) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S = R-star alpha G be the associated partial skew groupoid ring. We show that if a 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 is an element of mor(G). We use this result to prove that if a 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 is an element of 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.

  • 56.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.
    Pinedo, Héctor
    Universidad Industrial de Santander, Escuela de Matemáticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia.
    Epsilon-strongly graded rings, separability and semisimplicity2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, p. 1-24Article in journal (Refereed)
    Abstract [en]

    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 Nǎstǎsescu, 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.

  • 57.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.
    Richter, Johan
    Mälardalen University, Academy of Education, Culture and Communication,Västerås, Sweden.
    Non-associative Ore extensions2018In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 224, no 1, p. 263-292Article in journal (Refereed)
    Abstract [en]

    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.

  • 58.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.
    Richter, Johan
    Mälardalen University, Academy of Education, Culture and Communication,Box 883, Västerås, SE-72123, Sweden.
    Simplicity of Ore monoid rings2019In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 530, p. 69-85Article in journal (Refereed)
    Abstract [en]

    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.

  • 59.
    Patrik, Lundström
    University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences. University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Galois Module Structure of Field Extensions2007In: International Electronic Journal of Algebra, E-ISSN 1306-6048, Vol. 2, no 8, p. 100-105Article in journal (Refereed)
    Abstract [en]

    We show, in two different ways, that every finite field extension has a basis with the property that the Galois group of the extension acts faithfully on it. We use this to prove a Galois correspondence theorem for general finite field extensions. We also show that if the characteristic of the base field is different from two and the field extension has a normal closure of odd degree, then the extension has a self-dual basis upon which the Galois group acts faithfully.

12 51 - 59 of 59
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