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(ANTI)COMMUTATIVE ALGEBRAS WITH A MULTIPLICATIVE BASIS
Published online by Cambridge University Press: 15 December 2014
Abstract
A basis ${\mathcal{B}}=\{u_{i}\}_{i\in I}$ of a commutative or anticommutative algebra
$\mathfrak{C},$ over an arbitrary base field
$\mathbb{F}$, is called multiplicative if for any
$i,j\in I$ we have that
$u_{i}u_{j}\in \mathbb{F}u_{k}$ for some
$k\in I$. We show that if a commutative or anticommutative algebra
$\mathfrak{C}$ admits a multiplicative basis then it decomposes as the direct sum
$\mathfrak{C}=\bigoplus _{j}\mathfrak{i}_{j}$ of well-described ideals each one of which admits a multiplicative basis. Also the minimality of
$\mathfrak{C}$ is characterised in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is indexed by the family of its minimal ideals admitting a multiplicative basis.
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- Research Article
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- Copyright © 2014 Australian Mathematical Publishing Association Inc.
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