Локальная конечная базируемость и локальная представимость многообразий ассоциативных колец

Белов, А.Я.

Статья опубликована в журнале из списка RSCI Web of Science

Информация о цитировании статьи получена из Web of Science, Scopus
Статья опубликована в журнале из перечня ВАК
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 18 ноября 2015 г.

Автор: Белов А.Я.
Журнал: Доклады Академии наук
Том: 432
Номер: 6
Год издания: 2010
Издательство: ФГБУ "Издательство "Наука"
Местоположение издательства: Москва
Первая страница: 727
Последняя страница: 731
DOI: 10.1134/S1064562410030336
Аннотация: The paper under review announces important results on algebras with polynomial identities over arbitrary commutative Noetherian rings. The complete proofs have been recently published in a paper under the same title [Izv. Math. 74, No. 1, 1-126 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 1, 3-134 (2010; Zbl 1208.16022)]. Although the detailed version is well written and more informative for the experts, it is difficult to read due to the amount of the material and the numerous new ideas. The advantage of the present version is that it gives in short a sufficiently good account on the main results and some idea of the methods. The first topic of the paper is the Specht problem. The famous result of Kemer gives that every PI-algebra in characteristic 0 has a finite basis of its polynomial identities. In positive characteristic there are counterexamples, the first one due to the author of the present paper. Now the author proves that the Specht property holds locally, for finitely generated PI-algebras over commutative Noetherian rings. In characteristic 0, the Specht property is intimately related with the representability of finitely generated relatively free algebras. Now the author establishes an analogue of this result. If the ideal of the coefficients of the polynomial identities (in the underlying commutative ring) contains 1, then finitely generated relatively free algebras are representable. Here representability is understood in a more general sense than in the case of algebras over a field. It means that the algebra can be embedded into an algebra which is a finitely generated module over a commutative Noetherian ring. Additionally, relatively free algebras are residually finite. Finally, the author establishes that there exists an algorithm which decides whether a given polynomial identity is a consequence of a given finite system of polynomial identities.Reviewer: Vesselin Drensky (Sofia)
Добавил в систему: Белов Алексей Яковлевич

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Локальная конечная базируемость и локальная представимость многообразий ассоциативных колецстатья

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