Almost commutative varieties of associative rings and algebras over a finite field. (Q2342109)

A variety of associative algebras (or of semigroups) is said to be permutative if it satisfies an identity of the form \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\) for some permutation \(\sigma\) of \(\{1,2,\ldots,n\}\). The study of such varieties of semigroups started with \textit{M. Yamada} and \textit{N. Kimura} [Proc. Japan Acad. 34, 110-112 (1958; Zbl 0081.25501)] and for associative algebras with \textit{V. N. Latyshev} [Algebra Logika 8, 660-673 (1969; Zbl 0214.29402); translation in Algebra Logic 8, No. 6, 374-382 (1969)]. The purpose of the paper under review is to describe the minimal varieties which are not permutative, the so called almost permutative varieties, because they are the main obstacle for a variety to be permutative. In two previous papers, [\textit{O. B. Paison}, Russ. Math. 39, No. 1, 65-73 (1995); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1995, No. 1(392), 71-79 (1995; Zbl 0846.16018); and \textit{O. B. Finogenova}, Algebra Logic 51, No. 6, 519-534 (2013); translation from Algebra Logika 51, No. 6, 783-804 (2012; Zbl 1282.16025)], the author described, respectively, all finite dimensional algebras over finite fields which generate almost permutative varieties and all almost permutative varieties over infinite fields. The main result of the present paper presents a list of all almost permutative varieties over finite fields which are not generated by a finite algebra. The description is given both in the language of defining polynomial identities and generating algebras. Finally, the author characterizes the almost permutative varieties of rings. Reviewer's remark. The meaning of the notion of almost commutative varieties of algebras (as in the title of the English translation of the paper) does not reflect correctly the properties of the objects considered in the paper. The correct notion is almost permutative varieties.