Juan C. Gutierrez Fernandez

Commutative Finite-Dimensional Algebras Satisfying x(x(xy)) = 0 are Nilpotent

We investigate the structure of commutative non-associative algebras satisfying the identity x(x(xy)) = 0. Recently, Correa and Hentzel proved that every commutative algebra satisfying above identity over a field of characteristic ≠ 2 is solvable. We prove that every commutative finite-dimensional algebra 𝔄 over a field F of characteristic ≠ 2, 3 which satisfies the identity x(x(xy)) = 0 is nilpotent. Furthermore, we obtain new identities and properties for this class of algebras.

On Power-Associative Nilalgebras of Dimension N and Nilindex N-1

Whether or not a finite-dimensional, commutative, power-associative nilalgebra is solvable is a well-known open problem. In this paper, we describe commutative, power-associative nilalgebras of dimension n ≥ 6 and nilindex n − 1 based on the condition that n − 4 ≤ dim 𝔄3 ≤ n − 3. This paper is a continuation of [10], where we describe commutative power-associative nilalgebras of dimension and nilindex n. We observe that the Jordan case was obtained by L. Elgueta and A. Suazo in [2].

On Jordan-Nilalgebras of Index 3

We study commutative algebras that satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a new bound of the index of nilpotency. We prove that every commutative nilalgebra of nilindex 3 generated by k elements over a field of characteristic ≠ 2, 3 is nilpotent of index less than or equal to k + 5.

COMMUTATIVE POWER-ASSOCIATIVE ALGEBRAS OF NILINDEX FOUR

The aim of this article is to study the structure of the modules over a trivial algebra of dimension two in the variety ℳ of commutative and power-associative algebras. In particular, we classify the irreducible modules. These results enables us to understand better the structure of finite-dimensional power-associative algebras of nilindex 4.