Almost Commutative Varieties of Associative Rings and Algebras Over a Finite Field

Abstract

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 1 x 2⋯x n = x 1σ x 2σ ⋯x nσ , where σ is a nontrivial permutation on a set {1, 2, … , 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 [5], such varieties were characterized for the case of algebras over an infinite field.