On Some Extremal Varieties of Associative Algebras

Suppose that $F$ is a field of prime characteristic $p$ and $\mathbf V_p$ is the variety of associative algebras over $F$ defined by the identities $[[x,y],z]=0$ and $x^p=0$ if $p>2$ and by the identities $[[x,y],z]=0$ and $x^4=0$ if $p=2$ (here $[x,y]=xy-yx$). As is known, the free algebras of countable rank of the varieties $\mathbf V_p$ contain non-finitely generated $T$-spaces. We prove that the varieties $\mathbf V_p$ are minimal with respect to this property.