Dimensionally Nilpotent Jordan Algebras

Abstract

An algebra $A$ of dimension $n$ is called dimensionally nilpotent if it has a nilpotent derivation $\partial$ with the property that ${\partial ^{n - 1}} \ne 0$. Here we show that a dimensionally nilpotent Jordan algebra $A$ over a perfect field of characteristic not 2 or 3 is either (i) nilpotent, or (ii) one-dimensional modulo its maximal nilpotent ideal. This result is also extended to noncommutative Jordan algebras.