On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and 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 V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.