Some Results on the Structure of Finite Nilpotent Algebras Over Fields of Prime Characteristic

Abstract

Let $M$ be a finite dimensional commutative nilpotent algebra over a field $K$ of prime characteristic $p$. It has been conjectured that $\dim M\ge p\dim M^{(p)}$,
where $M^{(p)}$ is the subalgebra of $M$ generated by $x^p$, $x\in M$, $[2]$.
This was proved (by Eggert) in the case $\dim M^{(p)}\le 2$ in $1971$. This result was extended to the noncommutative case in $1994$ $[8]$. Not only is this conjecture important in its own right, but it was shown (by Eggert) that a proof of the above conjecture would result in a complete classification of the group of units of finite commutative rings of characteristic $p$ with an identity. In this short paper, we obtain a proof of Eggert’s conjecture in the case $\dim M^{(p)}=3$.