Associative Algebras Admitting a Quasi-multiplicative Basis

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.