On power-associative modules

The aim of this paper is to study the structure of irreducible modules in the variety M of commutative power-associative nilalgebras of nilindex <= 4. If A is an element of M with dimension at most 5, then we prove that A(2) is contained in the annihilator of every irreducible A-module in the variety M. Also, we consider the enveloping algebra of an algebra A in the variety M and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9.