Almost commutative varieties of associative rings and algebras over a finite field

Associative algebras over an associative commutative ring with unity are considered. A variety of algebras is said to be permutative if it satisfies an identity of the form
$$ x_1x_2\cdots x_n=x_{1\sigma}x_{2\sigma}\cdots x_{n\sigma}, $$
where $\sigma$ is a nontrivial permutation on a set $\{1,2,\dots,n\}$. Minimal elements in the lattice of all nonpermutative varieties are called almost permutative varieties. By Zorn's lemma, every nonpermutative variety contains an almost permutative variety as a subvariety. We describe almost permutative varieties of algebras over a finite field and almost commutative varieties of rings. In [Algebra Logika, 51, No. 6, 783–804 (2012)], such varieties were characterized for the case of algebras over an infinite field.