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(ANTI)COMMUTATIVE ALGEBRAS WITH A MULTIPLICATIVE BASIS

Part of: Basic linear algebra General nonassociative rings Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  15 December 2014

ANTONIO J. CALDERÓN MARTÍN
ANTONIO J. CALDERÓN MARTÍN*
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Cádiz, Campus de Puerto Real, 11510 Puerto Real, Cádiz, Spain email [email protected]
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Abstract

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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.

Keywords

commutative algebraanticommutative algebramultiplicative basisinfinite-dimensional algebrastructure theory

MSC classification

Primary: 17A60: Structure theory
Secondary: 15A03: Vector spaces, linear dependence, rank 17B05: Structure theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 211 - 218
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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