Nilpotence of a class of commutative power-associative nilalgebras

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 2t−3), then the right multiplication Rx is nilpotent and Rx2t−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 (A2)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.