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1.

Lundström, Patrik

University West, Department of Technology.

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.

2.

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.

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

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.

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.