Permutative pseudovarieties of associative algebras over a finite field (Q1910275)

An associative ring is called permutative if it satisfies an identity of the kind \(x_1\cdots x_k = x_{1\sigma}\cdots x_{k\sigma}\), where \(\sigma\) is a nontrivial permutation of the set \(\{1,2,\dots,k\}\). A pseudovariety is said to be permutative if it consists of permutative rings. The main result of the paper (Theorem 1) characterizes permutative pseudovarieties of associative algebras over a finite field by listing their ``forbidden objects''. This means that, for any fixed finite field \(F\), the author exhibits a list consisting of 2 infinite series of finite \(F\)-algebras and 2 ``sporadic'' finite \(F\)-algebras such that a pseudovariety of \(F\)-algebras is permutative if and only if it contains none of the algebras in the list. The characterization implies in particular that an \(n\)-element \(F\)-algebra is permutative if and only if it satisfies the identity \(x^{n! + 1} y^{n! + 1} = x^{n!} yxy^{n!}\) (Theorem 2) and if and only if all its 2-generated subalgebras are permutative. An example shows that, in contrast, an infinite algebra may fail to be permutative even in the case when all its finitely generated subalgebras are permutative.