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  • 1.
    Björkholdt, Elise
    et al.
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Generalizations of the normal basis theorem2004Ingår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 94, s. 5s. 185-190Artikel i tidskrift (Refereegranskat)
  • 2.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Brauer group rings2004Ingår i: JP Journal of Algebra, Number Theory & Applications, Vol. 4, s. 22s. 465-494Artikel i tidskrift (Refereegranskat)
  • 3.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Cohomology and Self-dual Normal Bases for Infinite Galois Field Extensions2002Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 256, s. 531-541Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We generalize an injectivity result obtained by Bayer-Fluckiger and Lenstra concerning pointed cohomology sets, defined by norm-one groups of finite-dimensional algebras with involution over fields k of characteristic different from 2, to the case of inverse limits of finite-dimensional k-algebras with involution. We use this generalization to obtain a result about self-dual normal bases for infinite Galois field extensions.

  • 4.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap, Avd för matematik och naturvetenskap.
    Cohomology and the normal basis theorem2007Ingår i: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 18, s. 69-72Artikel i tidskrift (Refereegranskat)
  • 5.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Crossed product algebras defined by separable extensions2005Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 283, s. 14s. 723-737Artikel i tidskrift (Refereegranskat)
  • 6.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Fuzzified categories of composition graphs2005Rapport (Övrig (populärvetenskap, debatt, mm))
    Abstract [en]

    Serving as a generalization of many examples of fuzzy algebraical systems equipped with a binary operation, we introduce fuzzy composition graphs and show that categories formed by such graphs are, in the sense of Wyler [10], top categories. By using this, we investigate projective and injective objects in such categories, and we determine when various limits and colimits, such as terminal and initial objects, products, coproducts, pullbacks, pushouts, equalizers, coequalizers, kernels and cokernels, exist in categories of this type and what they look like. These results are then applied to the categories of fuzzy sets, fuzzy categories, fuzzy groupoids, fuzzy monoids, fuzzy groups and fuzzy abelian groups.

  • 7.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap, Avd för matematik och naturvetenskap.
    Galois Module Structure of Field Extensions2007Ingår i: International Electronic Journal of Algebra, ISSN 1306-6048, E-ISSN 1306-6048, Vol. 2, nr 8, s. 100-105Artikel i tidskrift (Refereegranskat)
    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.

  • 8.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Generalized Brauer Algebras2002Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 30, nr 5, s. 2229-2270Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Using some ideas of Brauer, we introduce what we call generalized Brauer algebras and, as a special case, Brauer orders. We show that many well-known classes of so-called crossed product algebras, and in particular, the well-known crossed product orders, can be obtained as special instances of our construction. We prove several results showing when Brauer orders are Azumaya, maximal, hereditary or Gorenstein.

  • 9.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för naturvetenskap, lantmäteri- och maskinteknik.
    Good Magma Gradings On Rings2014Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 42, nr 12, s. 5357-5373Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Suppose that G and H are magmas and that R is a strongly G-graded ring. We show that there is a bijection between the set of good (zero) H-gradings of R and the set of (zero) magma homomorphisms from G to H. Thereby we generalize a result by Dascalescu, Ion, Nastasescu and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of good (zero) H-filters on R and the preordered set of (zero) submagmas of G \times H. These results are applied to category graded rings and, in particular, to the case when G and H are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of good H-gradings on R.

  • 10.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Hilbert 90 for Rings with ConjugationManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We show a version of Hilbert 90 that is valid for a large class ofalgebras many of which are not commutative, distributive orassociative. This class contains the n:th iteration of theConway-Smith doubling procedure. We use our version of Hilbert 90 toparametrize all solutions in ordered fields to the norm one equation for such algebras.

  • 11.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Hilbert 90 for algebras with conjugation2012Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 15, nr 1, s. 119-135Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show a version of Hilbert 90 that is valid for a large class ofalgebras many of which are not commutative, distributive orassociative. This class contains the n:th iteration of theConway-Smith doubling procedure. We use our version of Hilbert 90 toparametrize all solutions in ordered fields to the norm one equation for such algebras.

  • 12.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Normal Bases for Infinite Galois Ring Extensions1999Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 79, s. 235-240Artikel i tidskrift (Refereegranskat)
  • 13.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU). Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Normal Integral Bases for Infinite Abelian Extensions2001Ingår i: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 100, s. 79-83Artikel i tidskrift (Refereegranskat)
  • 14.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Pythagoreiska tripplar på sex olika sätt2008Ingår i: Normat, ISSN 0801-3500, Vol. 56, nr 3, s. 111-119Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Six different ways of parametrising Pythagorean triplets are presented, from the elementary arguments supplied by the Greek, via trigonometry to Gaussian integers and applications of Hilbers 90th theorem.

  • 15.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Self-dual Normal Bases for Infinite Galois Field Extensions1998Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 26, s. 4331-4341Artikel i tidskrift (Refereegranskat)
  • 16.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Self-dual normal bases for infinite odd abelian Galois ring extensions2006Ingår i: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 123, nr 1, s. 1-8Artikel i tidskrift (Refereegranskat)
  • 17.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Self-dual Normal Integral Bases for Infinite Unramified Extensions2002Ingår i: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 97, nr 2, s. 350-367Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.

  • 18.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Separable groupoid rings2006Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, s. 13s. 3029-3041Artikel i tidskrift (Övrig (populärvetenskap, debatt, mm))
  • 19.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap, Avd för matematik och naturvetenskap.
    Separable Groupoid Rings2006Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, nr 8, s. 3029-3041Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity

  • 20.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    The category of groupoid graded modules2004Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 100, s. 15s. 195-211Artikel i tidskrift (Refereegranskat)
  • 21.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, s. 1-13Artikel i tidskrift (Refereegranskat)
  • 22.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, s. 1-13Artikel i tidskrift (Refereegranskat)
  • 23.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Von Neumann finite endomorphism rings2003Ingår i: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 14, s. 9s. 223-232Artikel i tidskrift (Refereegranskat)
  • 24.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Weak topological functors2008Ingår i: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, nr 3, s. 211-215Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if then the induced functor of Wyler’s top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces. A ! B is a topological functor and J is a category,AJ ! BJ is topological. These results are applied to a generalization

  • 25.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Öinert, Johan
    Commutativity and Ideals in Category Crossed Products2010Ingår i: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 59, nr 4, s. 338-346Artikel i tidskrift (Refereegranskat)
  • 26.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Öinert, Johan
    LTH.
    Miyashita Action in Strongly Groupoid Graded RingsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We determine the commutant of homogeneous subringsin strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the groupgraded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction,that given a finite groupoid $G$, equipped with a nonidentitymorphism t : d(t) -> c(t), there is a strongly G-graded ring R with the properties that each R_s, for s in G, is nonzero and R_t is a nonfree left R_c(t)-module.

  • 27.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    Öinert, Johan
    LTH.
    Miyashita Action in Strongly Groupoid Graded Rings2012Ingår i: International Electronic Journal of Algebra, ISSN 1306-6048, E-ISSN 1306-6048, Vol. 11, s. 46-63Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We determine the commutant of homogeneous subrings in strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the groupgraded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction,that given a finite groupoid $G$, equipped with a nonidentitymorphism t : d(t) -> c(t), there is a strongly G-graded ring R with the properties that each R_s, for s in G, is nonzero and R_t is a nonfree left R_c(t)-module.

  • 28.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för elektro, lantmäteri och naturvetenskap.
    Öinert, Johan
    Köpenhamns Universitet.
    Skew category algebras associated with partially defined dynamical systems2012Ingår i: International Journal of Mathematics, ISSN 0129-167X, Vol. 23, nr 4, s. 16-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Top^op and show that it defines what we call a skew category algebra AxG. We study the connection between topological freeness of s and, on the one hand, ideal properties of AxG and, on the other hand, maximal commutativity of A in AxG. In particular, we show that if G is a groupoid and for each e in ob(G) the group of all morphisms from e to e is countable and the topological space s(e) is Tychonoff and Baire, then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, that is if I is a nonzero ideal of AxG, then I \cap A is not equal to zero; (iii) the ring A is a maximal abelian complex subalgebra of AxG. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

  • 29.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Öinert, Johan
    LTH.
    The Ideal Intersection Property for Groupoid Graded Rings2012Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 40, nr 5, s. 1860-1871Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that if a groupoid graded ring hasa certain nonzero ideal property, then the commutant of the center of the principal component of the ringhas the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras withcommutative principal component, the principal componentis maximal commutative if and only if it has the ideal intersection property.

  • 30.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för elektro, lantmäteri och naturvetenskap.
    A combinatorial proof of associativity of Ore extensions2013Ingår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 313, nr 23, s. 2748-2750Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use a counting argument to show that Ore extensions are associative. 

  • 31.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för naturvetenskap, lantmäteri- och maskinteknik.
    A Proof of the Cosine Addition Formula Using the Law of Cosines2014Ingår i: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 87, nr 2, s. 144-144Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a proof of the cosine addition formula using the law of cosines.

  • 32.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    A proof of the law of sines using the law of cosines2017Ingår i: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 90, nr 3, s. 180-181Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a proof of the law of sines using the law of cosines. © Mathematical Association of America.

  • 33.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för maskinteknik och naturvetenskap.
    Fuzzy crossed product algebras2015Ingår i: Annals of Fuzzy Mathematics and Informatics, ISSN 2093-9310, E-ISSN 2287-6235, Vol. 10, nr 6, s. 959-969Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce fuzzy groupoid graded rings and, as a par-ticular case, fuzzy crossed product algebras. We show that there is abijection between the set of fuzzy graded is omorphism equivalence classes of fuzzy crossed product algebras and the associated second cohomology group. This generalizes a classical result for crossed product algebras to thefuzzy situation. Thereby, we quantize the difference of richness between the fuzzy and the crisp case. We give several examples showing that in the fuzzy case the associated second cohomology group is much ner than in the classical situation. In particular, we show that the cohomology group may by in nite in the fuzzy case even though it is trivial in the crisp case.

  • 34.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    Limits of Fuzzy Categories2012Ingår i: Advances in Fuzzy Systems, ISSN 1687-7101, E-ISSN 1687-711X, Vol. 13, nr 2, s. 77-96Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Serving as a generalization of many examples of fuzzy algebraical systems, we introduce fuzzy categories and show that categories formed by fuzzy categories are topological. By using this, we show results concerning the existence of limits and colimits in such categories. We apply these results to the categories of fuzzy sets, fuzzy categories, fuzzy groupoids, fuzzy monoids, fuzzy groups, fuzzy abelian groups and fuzzy ordered sets. Thereafter, we determine the complete ordered lattice structure of the collection of grade maps on some finite categories, in particular on cyclic groups of prime power order. We use this in the end of the article to construct grade maps on p-adic groups.

  • 35.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Partial category actions on sets and topological spaces2018Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, nr 2, s. 671-683Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schroder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.

  • 36.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för maskinteknik och naturvetenskap.
    Oinert, Johan
    Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden.
    Simple semigroup graded rings2015Ingår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 14, nr 7Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers’ on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves’. We also point out how Jespers’ result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

  • 37.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för elektro, lantmäteri och naturvetenskap.
    Öinert, J.
    Lund University, Centre for Mathematical Sciences.
    Simple rings and degree maps2014Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 401, s. 201-219Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying A I ⊆ I A. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-negative integers satisfying the following two conditions: (d1) if a ∈ A, then d(a) = 0 if and only if a = 0; (d2) there is a subset X of B generating B as a ring such that for each non-zero ideal I of A and each non-zero a ∈ I there is a non-zero a ' ∈ I with d(a ') ≤ d(a) and d(a 'b - ba ') < d(a) for all b ∈ X. We show that if the centralizer C of B in A is an A-simple ring, every intersection of C with an ideal of A is A-invariant, A C A = A and there is a degree map for A/B, then A is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings. © 2013 Elsevier Inc.

  • 38.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.
    Group gradations on Leavitt path algebras2019Ingår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 39.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för maskinteknik och naturvetenskap.
    Öinert, Johan
    Lund University, Centre for Mathematical Sciences.
    Outer Partial Actions and Partial Skew Group Rings2015Ingår i: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 67, nr 5, s. 1144-1160Artikel i tidskrift (Refereegranskat)
    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.

  • 40.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för elektro, lantmäteri och naturvetenskap.
    Öinert, Johan
    University of Copenhagen, Department of Mathematical Sciences.
    Simple skew category algebras associated with minimal partially defined dynamical systems2013Ingår i: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 33, nr 9, s. 4157-4171Artikel i tidskrift (Refereegranskat)
    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.

  • 41.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    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 rings2018Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, s. 433-452Artikel i tidskrift (Refereegranskat)
    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.

  • 42.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    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 semisimplicity2018Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, s. 1-24Artikel i tidskrift (Refereegranskat)
    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.

  • 43.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    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 extensions2018Ingår i: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 224, nr 1, s. 263-292Artikel i tidskrift (Refereegranskat)
    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.

  • 44.
    Nystedt, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    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 rings2019Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 530, s. 69-85Artikel i tidskrift (Refereegranskat)
    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.

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