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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