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Algebras having bases that consist solely of units

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Abstract

Algebras having bases that consist entirely of units (called invertible algebras) are studied. Among other results, it is shown that all finite-dimensional algebras over a field other than the binary field F 2 have this property. Invertible finite-dimensional algebras over F 2 are fully characterized. Examples of invertible algebras are shown to include all (non-trivial) matrix rings over arbitrary algebras. In addition, various families of algebras, including group rings and crossed products, are characterized in terms of invertibility. Invertibility of infinite-dimensional algebras is also explored.

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Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Sergio R. López-Permouth & Nick Pilewski

  2. Department of Mathematics, Otterbein University, Westerville, OH, 43081, USA

    Jeremy Moore

  3. Department of Mathematics, Eastern Kentucky University, Richmond, KY, 40475, USA

    Steve Szabo

Authors
  1. Sergio R. López-Permouth
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  2. Jeremy Moore
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  3. Nick Pilewski
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  4. Steve Szabo
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Correspondence to Sergio R. López-Permouth.

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López-Permouth, S.R., Moore, J., Pilewski, N. et al. Algebras having bases that consist solely of units. Isr. J. Math. 208, 461–482 (2015). https://doi.org/10.1007/s11856-015-1208-2

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  • DOI: https://doi.org/10.1007/s11856-015-1208-2

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