Characterization of non-matrix varieties of associative algebras. (Q532618)

Polynomial identities of matrices are used to measure the complexity of T-ideals and the corresponding varieties of associative algebras. Non-matrix varieties, i.e., varieties which do not contain the \(2\times 2\) matrix algebra, are considered to have simple structure from this point of view and have been an object of intensive study in the last 50 years. There are several known characterizations of non-matrix varieties. In the paper under review the authors give some new characterizations in terms of properties of nilelements. Let \(\mathcal V\) be a variety of associative algebras over an infinite field. Then the following conditions are equivalent: (1) The variety \(\mathcal V\) is non-matrix; (2) Any finitely generated algebra \(A\in\mathcal V\) satisfies an identity of the form \([x_1,x_2]\cdots[x_{2s-1},x_{2s}]\equiv 0\); (3) If \(A\in\mathcal V\), then for any nilelements \(a,b\in A\), the element \(a+b\) is again a nilelement. The key roles in the theory of non-matrix varieties are played by the Grassmann algebra \(E\) in countable many generators and its tensor square \(E\otimes E\). The authors also give similar characterizations for non-matrix varieties over fields of characteristic zero that do not contain \(E\) or \(E\otimes E\).