DIMENSIONS OF NILPOTENT ALGEBRAS OVER FIELDS OF PRIME CHARACTERISTIC

In this short paper we consider the conjecture that for a finite dimen sional commutative nilpotent algebra M over a perfect field of prime c haracteristic 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 p rove that for any finite dimensional nilpotent algebra M (not necessar ily commutative) over any field of prime characteristic p, dim M great er than or equal to p dim M((p)) for dim M((p)) less than or equal to 2.