Dimensionally nilpotent Jordan algebras

Abstract

An algebra A A of dimension n n is called dimensionally nilpotent if it has a nilpotent derivation ∂ \partial with the property that ∂ n − 1 ≠ 0 {\partial ^{n - 1}} \ne 0 . Here we show that a dimensionally nilpotent Jordan algebra A 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.