Some results on the structure of a class of commutative power-associative nilalgebras of nilindex four

Abstract

When a commutative nilalgebra has nilidex four, we can separate their study in three big cases. One of them is when the algebra \({\mathcal {A}}\) satisfies the identity \( (((yx)x)x)x= 0\). In this paper we get results on the structure of this kind of algebras. Let W be the subspace of \({\mathcal {A}}\) generated by the cubes. We prove that \(W^2 = 0\) and we use this to get others properties of W.