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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. Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Generalizations of the normal basis theorem2004Inngår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 94, s. 5s. 185-190Artikkel i tidsskrift (Fagfellevurdert)
  • 2.
    Cala, Juan
    et al.
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Pinedo, Hector
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Graded modules over object-unital groupoid graded rings2021Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 50, nr 2, s. 444-462Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.). 

  • 3.
    Cala, Juan
    et al.
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Pinedo, Hector
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Object-unital groupoid graded rings, crossed products and separability2021Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, nr 4, s. 1676-1696Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 4.
    Darpö, Erik
    et al.
    Nagoya University, Graduate School of Mathematics, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan (JPN).
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras2021Inngår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 24, s. 1245-1258Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. Types of algebras that are studied include alternative, flexible, quadratic and involutive algebras, as well as algebras obtained by the Cayley–Dickson doubling process. Our results include precise criteria for von-Neumann finiteness and reversibility of involutive algebras in terms of isomorphism types of their 3-dimensional subalgebras. © 2020, The Author(s).

  • 5.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Algebra, trigonometri och analys2023 (oppl. 1)Bok (Annet vitenskapelig)
  • 6.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Brauer group rings2004Inngår i: JP Journal of Algebra, Number Theory & Applications, Vol. 4, s. 22s. 465-494Artikkel i tidsskrift (Fagfellevurdert)
  • 7.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Cohomology and Self-dual Normal Bases for Infinite Galois Field Extensions2002Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 256, s. 531-541Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 8.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap, Avd för matematik och naturvetenskap.
    Cohomology and the normal basis theorem2007Inngår i: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 18, s. 69-72Artikkel i tidsskrift (Fagfellevurdert)
  • 9.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Crossed product algebras defined by separable extensions2005Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 283, s. 14s. 723-737Artikkel i tidsskrift (Fagfellevurdert)
  • 10.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Double Calculus2023Inngår i: Surveys in Mathematics and its Applications, ISSN 1843-7265, E-ISSN 1842-6298, Vol. 18, s. 27-48Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bögel [1, 2], already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes many two-variable extensions of classical results from single variable calculus, such as Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Fermat’s extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus. 

    Fulltekst (pdf)
    fulltext
  • 11.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Elementär optimeringslära2021Bok (Annet vitenskapelig)
    Abstract [sv]

    Elementär optimeringslära inleds med en repetition av grundläggande matematikkunskaper om algebra, ekvationer, matriser, funktioner och derivata. Därefter behandlas linjär optimering, först i två variabler med fokus på geometrisk förståelse och därefter, i det allmänna fallet, med simplexmetoden. Denna metod tillämpas i boken även inom det spännande matematikområdet spelteori. Boken redogör för olika metoder för optimering av allmänna tvåvariabelfunktioner, dels över kompakta områden, dels med Lagranges sats givet ett bivillkor och dels med Hessianen. Boken tar avslutningsvis upp optimering på grafer med Kruskals, Prims och Dijkstras algoritmer.

  • 12.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Fuzzified categories of composition graphs2005Rapport (Annet (populærvitenskap, 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.

    Fulltekst (pdf)
    FULLTEXT01
  • 13.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU). Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Generalized Brauer Algebras2002Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 30, nr 5, s. 2229-2270Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 14.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för naturvetenskap, lantmäteri- och maskinteknik.
    Good Magma Gradings On Rings2014Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 42, nr 12, s. 5357-5373Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 15.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Hilbert 90 for algebras with conjugation2012Inngår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 15, nr 1, s. 119-135Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 16.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Normal Bases for Infinite Galois Ring Extensions1999Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 79, s. 235-240Artikkel i tidsskrift (Fagfellevurdert)
  • 17.
    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 Extensions2001Inngår i: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 100, s. 79-83Artikkel i tidsskrift (Fagfellevurdert)
  • 18.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Primitives of continuous functions via polynomials2022Inngår i: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, s. 1-5Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In standard books on calculus the existence of primitive functions of continuous functions is proved, in one way or another, using Riemann sums. In this note we present a completely different self-contained, however probably folkloristic, proof of this existence. Our proof combines, on the one hand, the so-called Stone Weierstrass theorem on uniform approximation of continuous functions on the unit interval by polynomials, and, on the other hand, a classical result from calculus on the existence of limits of differentiated sequences of functions. The sought for primitive is then constructed as the limit of primitives of the polynomials approximating the original function.

  • 19.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Pythagoreiska tripplar på sex olika sätt2008Inngår i: Normat, ISSN 0801-3500, Vol. 56, nr 3, s. 111-119Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 20.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik (HTU).
    Self-dual Normal Bases for Infinite Galois Field Extensions1998Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 26, s. 4331-4341Artikkel i tidsskrift (Fagfellevurdert)
  • 21.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Self-dual normal bases for infinite odd abelian Galois ring extensions2006Inngår i: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 123, nr 1, s. 1-8Artikkel i tidsskrift (Fagfellevurdert)
  • 22.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Self-dual Normal Integral Bases for Infinite Unramified Extensions2002Inngår i: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 97, nr 2, s. 350-367Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 23.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap, Avd för matematik och naturvetenskap.
    Separable Groupoid Rings2006Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, nr 8, s. 3029-3041Artikkel i tidsskrift (Fagfellevurdert)
    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

  • 24.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Separable groupoid rings2006Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, s. 13s. 3029-3041Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
  • 25.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    The category of groupoid graded modules2004Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 100, s. 15s. 195-211Artikkel i tidsskrift (Fagfellevurdert)
  • 26.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, s. 1-13Artikkel i tidsskrift (Fagfellevurdert)
  • 27.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, s. 1-13Artikkel i tidsskrift (Fagfellevurdert)
  • 28.
    Lundström, Patrik
    Högskolan Väst, Institutionen för teknik, matematik och datavetenskap.
    Von Neumann finite endomorphism rings2003Inngår i: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 14, s. 9s. 223-232Artikkel i tidsskrift (Fagfellevurdert)
  • 29.
    Lundström, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Weak topological functors2008Inngår i: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, nr 3, s. 211-215Artikkel i tidsskrift (Fagfellevurdert)
    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

  • 30.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Oinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona (SWE).
    Group gradations on Leavitt path algebras (vol 19, 2050165, 2020)2023Inngår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, artikkel-id 2492001Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The results that are stated in P. Nystedt and J. Oinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

  • 31.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Öinert, Johan
    Commutativity and Ideals in Category Crossed Products2010Inngår i: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 59, nr 4, s. 338-346Artikkel i tidsskrift (Fagfellevurdert)
  • 32.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Öinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, Karlskrona (SWE).
    Corrigendum: Group gradations on Leavitt path algebras2023Inngår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829Artikkel i tidsskrift (Annet vitenskapelig)
    Abstract [en]

    The results that are stated in P. Nystedt and J. Öinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

  • 33.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    Öinert, Johan
    LTH.
    Miyashita Action in Strongly Groupoid Graded Rings2012Inngår i: International Electronic Journal of Algebra, E-ISSN 1306-6048, Vol. 11, s. 46-63Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 34.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Öinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona (SWE).
    Simplicity of Leavitt Path Algebras via Graded Ring Theory2023Inngår i: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 108, nr 3, s. 428-437Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Suppose that R is an associative unital ring and that E= (E-0, E-1, r, s) is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra L-R(E) is simple if and only if R is simple, E-0 has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

    Fulltekst (pdf)
    fulltext
  • 35.
    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 systems2012Inngår i: International Journal of Mathematics, ISSN 0129-167X, Vol. 23, nr 4, s. 16-Artikkel i tidsskrift (Fagfellevurdert)
    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.

    Fulltekst (pdf)
    fulltext
  • 36.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för lantmäteriteknik och matematik.
    Öinert, Johan
    LTH.
    The Ideal Intersection Property for Groupoid Graded Rings2012Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 40, nr 5, s. 1860-1871Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 37.
    Lundström, Patrik
    et al.
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences, Karlskrona (SWE).
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences, Karlskrona (SWE).
    Non-Unital Ore Extensions2023Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 172, nr 2, s. 217-229Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; delta], under the hy-pothesis that R is s-unital and ker(delta) contains a non-zero idempotent. This result gener-alizes a result by oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.

  • 38.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för elektro, lantmäteri och naturvetenskap.
    A combinatorial proof of associativity of Ore extensions2013Inngår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 313, nr 23, s. 2748-2750Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 39.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avd för naturvetenskap, lantmäteri- och maskinteknik.
    A Proof of the Cosine Addition Formula Using the Law of Cosines2014Inngår i: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 87, nr 2, s. 144-144Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 40.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    A proof of the law of sines using the law of cosines2017Inngår i: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 90, nr 3, s. 180-181Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

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

  • 41.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Arc length of function graphs via Taylor’s formula2021Inngår i: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 52, nr 2, s. 310-323Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We use Taylor’s formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of functions f with the property that (Formula presented.) has a primitive, including classical examples by Neile, van Heuraet and Fermat, as well as more recent ones induced by Pythagorean triples of functions. We also discuss potential benefits for our proposed definition of arc length in introductory calculus courses. © 2020, © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

  • 42.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för maskinteknik och naturvetenskap.
    Fuzzy crossed product algebras2015Inngår i: Annals of Fuzzy Mathematics and Informatics, ISSN 2093-9310, E-ISSN 2287-6235, Vol. 10, nr 6, s. 959-969Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 43.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap.
    Limits of Fuzzy Categories2012Inngår i: Advances in Fuzzy Systems, ISSN 1687-7101, E-ISSN 1687-711X, Vol. 13, nr 2, s. 77-96Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 44.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Noncommutatively Graded Algebras2020Inngår i: Algebraic Structures and Applications / [ed] Silvestrov S., Malyarenko A., Rančić M. (eds), Springer, Cham: Springer International Publishing , 2020, Vol. 317, s. 371-383Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning identity elements, inverses, existence of limits and colimits and adjointness of certain functors. In the particular instance of noncommutatively graded Lie algebras, we establish the existence of universal graded enveloping algebras and we show a graded version of the Poincaré-Birkhoff-Witt theorem.

  • 45.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Partial category actions on sets and topological spaces2018Inngår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, nr 2, s. 671-683Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 46.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Poisson’s fundamental theorem of calculus via Taylor’s formula2020Inngår i: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 51, nr 5, s. 799-805Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We use Taylor’s formula with Lagrange remainder to make a modern adaptation of Poisson’s proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

  • 47.
    Nystedt, Patrik
    Högskolan Väst, Institutionen för ingenjörsvetenskap, Avdelningen för Matematik, Data- och Lantmäteriteknik.
    Simplicity of algebras via epsilon-strong systems2020Inngår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 162, nr 2, s. 279-301Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We obtain sufficient criteria for simplicity of systems, that is, rings R that are equipped with a family of additive subgroups R-s for s is an element of S, where S is a semigroup satisfying R = Sigma (s is an element of S) R-s and RsRt subset of R-st for s, t is an element of S. These criteria are specialized to obtain sufficient criteria for simplicity of what we call s-unital epsilon-strong systems, that is, systems where S is an inverse semigroup, R is coherent, in the sense that R-s subset of R-t for all s, t is an element of S with s <= t and for each s is an element of S, the RsRs*-Rs*Rs -bimodule R-s is s-unital. As an application, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings by Beuter, Goncalves, Oinert and Royer, and then for Steinberg algebras over non-commutative rings by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michell.

  • 48.
    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 rings2015Inngår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 14, nr 7Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 49.
    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 maps2014Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 401, s. 201-219Artikkel i tidsskrift (Fagfellevurdert)
    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 ') &lt; 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.

  • 50.
    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 algebras2020Inngår i: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 19, nr 9, artikkel-id 2050165Artikkel i tidsskrift (Fagfellevurdert)
    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.

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