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

Let \(M\) be a finite dimensional commutative nilpotent algebra over a field \(K\) of prime characteristic \(p\). It has been conjectured that \(\dim M \geq p \;\dim M^{(p)}\),
where \(M^{(p)}\) is the subalgebra of \(M\) generated by \(x^p\), \(x \in M\), \([2]\).
This was proved (by Eggert) in the case \(\dim M^{(p)} \leq 2\) 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 \(\dim M^{(p)} = 3\).