Abstract
A basis \({\mathcal B}=\{e_{i}\}_{i \in I}\) of an associative algebra \({\frak A},\) over an arbitrary base field \({\mathbb F}\), is called multiplicative if for any i,j∈I we have that \(e_{i}e_{j} \in {\mathbb F} e_{k}\) for some k∈I. The class of associative algebras admitting a multiplicative basis can be seen as a particular case of the more general class of associative algebras admitting a quasi-multiplicative basis. In the present paper we prove that if an associative algebra \({\frak A}\) admits a quasi-multiplicative basis then it decomposes as the sum of well-described ideals admitting quasi-multiplicative bases plus (maybe) a certain linear subspace. Also the minimality of \({\frak A}\) is characterized in terms of the quasi-multiplicative basis and it is shown that, under mild conditions, the above decomposition is actually the direct sum of the family of its minimal ideals admitting a quasi-multiplicative basis.
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The author would like to thank the referee for his exhaustive review of the paper as well as his suggestions which have helped to improve the work.
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Presented by: Raymundo Bautista.
Supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2010-15223.
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Calderon Martin, A.J. Associative Algebras Admitting a Quasi-multiplicative Basis. Algebr Represent Theor 17, 1889–1900 (2014). https://doi.org/10.1007/s10468-014-9482-y
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DOI: https://doi.org/10.1007/s10468-014-9482-y