A Class of Almost Commutative Nilalgebras

Abstract

The purpose of this paper is to investigate a class of nonassociative nilalgebras which have absolute zero divisors. If a nilalgebra is nilpotent, it, of course, possesses an absolute zero divisor. For the nilpotence of nonassociative nilalgebras, the situation however becomes quite complicated even in the finite-dimensional case. For example, Gerstenhaber [3] has conjectured the nilpotence of commutative nilalgebras. While Gerstenhaber and Myung [4] prove that any commutative nilalgebra of dimension ≦ 4 in characteristic ≠ 2 is nilpotent, Suttles [9] discovered an example of a 5-dimensional commutative nilalgebra which is solvable but not nilpotent.