Контрпримеры к проблеме Шпехта

What is it about?

The paper under review deals with the famous Specht problem of whether every variety $V$ of associative algebras is finitely based (i.e. whether all polynomial identities satisfied in $V$ follow from some finite system of such identities for $V$). A celebrated theorem of A. R. Kemer [Ideals of identities of associative algebras, Translated from the Russian by C. W. Kohls, Amer. Math. Soc., Providence, RI, 1991; MR1108620] gave an affirmative answer to this problem when the base field is of characteristic 0. The paper under review starts with a survey of the known results in this direction. Let $K\langle X\rangle$ be the free associative algebra freely generated by the set $X$ of variables over a field $K$. If the set $X$ is finite, one has the so-called local Specht problem of whether every ideal of identities ($T$-ideal) in $K\langle X\rangle$ is finitely based as $T$-ideal. The local Specht problem was resolved positively by Kemer for infinite fields, and by the author of the paper under review for finite fields. An important step in the global case was made by A. Grishin who gave an example of an infinitely based $T$-space over a field of characteristic 2. (Recall that a vector subspace of $K\langle X\rangle$ is a $T$-space if it is closed under all endomorphisms of $K\langle X\rangle$.) Later examples of infinitely based $T$-spaces in any positive characteristic were found as well. The main result of the paper under review is the negative solution of the (global) Specht problem in the case of positive characteristic. The author constructs a sequence of polynomials that generate an infinitely based $T$-ideal in $K\langle X\rangle$ where $K$ is a field of positive characteristic. Reviewed by Plamen Koshlukov

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