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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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