Контрпримеры к проблеме Шпехта

Белов, А.Я.

Автор: Белов А.Я.
Журнал: Математический сборник
Том: 191
Номер: 3
Год издания: 2000
Издательство: МИАН
Местоположение издательства: Москва
Первая страница: 13
Последняя страница: 24
DOI: 10.4213/sm460
Аннотация: One of the central problems of the theory of varieties of (any universal) algebras is the finite basis (or Specht) problem: Can any variety be defined by a finite system of identical relations? For varieties of associative rings (or algebras over a field) the Specht problem can be restated as: Is any T-ideal of the free associative ring (algebra) on a countable set of free generators, finitely generated as a T-ideal? For many important classes of algebraic systems there exist examples of infinitely based varieties (e.g. for groups, semigroups, Lie algebras over a field of positive characteristic, etc.). For associative algebras over a field of characteristic 0 the deep structure theory of T-ideals developed by Kemer allowed him to obtain the positive solution to the Specht problem.\par Up till now, the Specht problem was still open for associative algebras over a field of positive characteristic and the paper under review contains a nice counterexample. The author shows that the variety of associative algebras over a field of odd characteristic $p$ defined, for a fixed $q=p^s$, $s>1$, by the polynomial identities $$R_n=[[u,t],t]\prod_{i=1}^n\left(x_i^{p-1}y_i^{p-1}[x_i,y_i]\right)([t,[t,v]][[u,t],t])^{q-1}[t,[t,v]]=0$$ is not finitely based. In the case of characteristic 2 the given system of polynomial identities is even simpler. Although intricated, the proof is based on simple ideas. It combines methods already used to construct counterexamples for groups and Lie algebras with the notion of T-space recently introduced by Grishin and further developed by {\it V. V. Shchigolev} [Sb. Math. 191, No. 3, 459-476 (2000); translation from Mat. Sb. 191, No. 3, 143-160 (2000; see the review Zbl 0960.16031 below)]. Recently {\it A. V. Grishin} [On non-Spechtianness of the variety of associative rings that satisfy the identity $x^{32}=0$. Electron. Res. Announc. Am. Math. Soc. 6, No. 7, 50-51 (2000) (electronic)] has announced another counterexample to the Specht problem for associative algebras over a field of characteristic 2. In particular, his algebras are nil of index 32.Reviewer: Vesselin Drensky (Sofia
Добавил в систему: Белов Алексей Яковлевич

	ИСТИНА	Войти в систему Регистрация
	ИСТИНА ЦЭМИ РАН
	Главная Поиск Статистика О проекте Помощь

ИСТИНА

ИСТИНА ЦЭМИ РАН

Контрпримеры к проблеме Шпехтастатья

Прикрепленные файлы