Nilpotence of a class of commutative power-associative nilalgebras

Abstract

Let A be a commutative algebra over a field F of characteristic ≠ 2 , 3 . In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21–31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2 t − 3 ), then the right multiplication R x is nilpotent and R x 2 t − 3 = 0 for all x ∈ A . In this work, we prove that this result is also valid for commutative power-associative algebras of characteristic ⩾ t. In Section 3, we prove that when A is a power-associative nilalgebra of dimension ⩽6, then A is nilpotent or ( A 2 ) 2 = 0 . In Section 4, we prove that every power-associative nilalgebra A of dimension n and nilindex t ⩾ n − 1 is either nilpotent of index t or isomorphic to the Suttles' example.