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  • 1.
    Nystedt, Patrik
    et al.
    University West, Sweden.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Simple graded rings, non-associative crossed products and Cayley-Dickson doublingsManuscript (preprint) (Other academic)
    Abstract [en]

    We show that if a non-associative 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 from the associative case to the non-associative situation. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply this result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.

  • 2.
    Nystedt, Patrik
    et al.
    University West, SWE.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Pinedo, Héctor
    Industrial University of Santander, COL.
    Epsilon-strongly graded rings, separability and semisimplicity2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, no Nov., 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 collection of strongly graded rings and the family of unital partial crossed products. We determine when epsilon-strongly graded rings are separable over their principal components. Thereby, we simultaneously generalize a result for strongly group-graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the family of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the family 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 Simon concerning when graded rings can be presented as partial crossed products.

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