On power-associative modules

Abstract

The aim of this paper is to study the structure of irreducible modules in the variety [Formula: see text] of commutative power-associative nilalgebras of nilindex [Formula: see text]. If [Formula: see text] with dimension at most 5, then we prove that [Formula: see text] is contained in the annihilator of every irreducible [Formula: see text]-module in the variety [Formula: see text]. Also, we consider the enveloping algebra of an algebra [Formula: see text] in the variety [Formula: see text] and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9.