О многообразиях, порожденных кольцом, конечномерным над центроидом

Белов, А.Я.

Автор: Белов А.Я.
Журнал: Успехи математических наук
Том: 62
Номер: 2(374)
Год издания: 2007
Издательство: ФГБУ "Издательство "Наука"
Местоположение издательства: Москва
Первая страница: 171
Последняя страница: 172
DOI: 10.4213/rm6805
Аннотация: It is known that varieties of associative algebras over an infinite field are locally generated by a finite-dimensional algebra [{\it A. R. Kemer}, Math. USSR, Izv. 37, No. 1, 69-96 (1991); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 4, 726-753 (1990; Zbl 0784.16016)]. Nevertheless, this is false already for algebras over a finite field. The simplest example is the variety generated by the semidirect product of a finite field k and the polynomial ring k[x]. A more interesting example is the variety generated by matrices of the form \left(\smallmatrix P&*\\ 0&P^q\endsmallmatrix\right), where P∈k[x] and the field k satisfies the identity zq=z. However, any variety of associative algebras over an associative-commutative ring Φ is generated by a subalgebra of an algebra which is finite-dimensional over its center [{\it A. Ya. Belov}, D.Sc. thesis, Moscow State Univ., Moscow (2002)]. The main theorem in this article is as follows. Theorem 1. The ideal of identities of a variety \germM consisting of algebras of arbitrary signature and generated by an algebra that is finite-dimensional over its centroid, is the intersection of the ideal of identities of a finite algebra and the ideal of identities of a homogeneous variety. If the variety \germM is unitarily closed, then it is generated by an algebra that is finite-dimensional over its center.
Добавил в систему: Белов Алексей Яковлевич

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О многообразиях, порожденных кольцом, конечномерным над центроидомстатья

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