n-Algebras admitting a multiplicative basis

Abstract

Let [Formula: see text] be an [Formula: see text]-algebra of arbitrary dimension and over an arbitrary base field [Formula: see text]. A basis [Formula: see text] of [Formula: see text] is said to be multiplicative if for any [Formula: see text], we have either [Formula: see text] or [Formula: see text] for some (unique) [Formula: see text]. If [Formula: see text], we are dealing with algebras admitting a multiplicative basis while if [Formula: see text] we are speaking about triple systems with multiplicative bases. We show that if [Formula: see text] admits a multiplicative basis then it decomposes as the orthogonal direct sum [Formula: see text] of well-described ideals admitting each one a multiplicative basis. Also, the minimality of [Formula: see text] is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.