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  • 1.
    Cala, Juan
    et al.
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Pinedo, Hector
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Graded modules over object-unital groupoid graded rings2021In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 50, no 2, p. 444-462Article in journal (Refereed)
    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.). 

  • 2.
    Cala, Juan
    et al.
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Pinedo, Hector
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Object-unital groupoid graded rings, crossed products and separability2021In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 4, p. 1676-1696Article in journal (Refereed)
    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.

  • 3.
    Lundström, Patrik
    University West, Department of Technology. University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Generalized Brauer Algebras2002In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 30, no 5, p. 2229-2270Article in journal (Refereed)
    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.

  • 4.
    Lundström, Patrik
    University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.
    Good Magma Gradings On Rings2014In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 42, no 12, p. 5357-5373Article in journal (Refereed)
    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.

  • 5.
    Lundström, Patrik
    University West, Department of Technology.
    Self-dual Normal Bases for Infinite Galois Field Extensions1998In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 26, p. 4331-4341Article in journal (Refereed)
  • 6.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences.
    Separable Groupoid Rings2006In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, no 8, p. 3029-3041Article in journal (Refereed)
    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

  • 7.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Separable groupoid rings2006In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, p. 13p. 3029-3041Article in journal (Other (popular science, discussion, etc.))
  • 8.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Öinert, Johan
    LTH.
    The Ideal Intersection Property for Groupoid Graded Rings2012In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 40, no 5, p. 1860-1871Article in journal (Refereed)
    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.

  • 9.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Partial category actions on sets and topological spaces2018In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 2, p. 671-683Article in journal (Refereed)
    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.

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