Dimensions of nilpotent algebras over fields of prime characteristic

Abstract

In this short paper we consider the conjecture that for a finite dimensional commutative nilpotent algebra M over a perfect field of prime characteristic p, dim M greater than or equal to p dim M((p)) where M(p) is the subalgebra of M generated by x(p), x is an element of M. We prove that for any finite dimensional nilpotent algebra M (not necessarily commutative) over any field of prime characteristic p, dim M greater than or equal to p dim M((p)) for dim M((p)) less than or equal to 2.