Локальная конечная базируемость и локальная представимость многообразий ассоциативных колец

What is it about?

Many important results in the theory of PI-algebras, established initially in characteristic 0, can be transferred in the case of PI-algebras in positive characteristic or even for PI-algebras over commutative Noetherian rings. Quite often this happens with additional difficulties and involving new ideas. The paper under review is an important typical example in this direction. It is devoted to the proof of several key results related with the Specht problem and representability of relatively free algebras. One of the central results of the theory of Kemer is that every PI-algebra over a field of characteristic 0 satisfies the Specht property, i.e., has a finite basis of its polynomial identities. This is intimately related with the representability of finitely generated relatively free algebras. Such an algebra can be embedded into a matrix algebra over an extension of the base field. In positive characteristic the author and a little bit later Grishin and Shchigolev constructed PI-algebras without finite bases of their polynomial identities. In this context one of the central results of the paper under review is that PI-algebras in positive characteristic satisfy the Specht property locally: Finitely generated PI-algebras satisfy the ascending chain condition for T-ideals. Additionally, finitely generated relatively free algebras are representable. These results have their generalization for PI-algebras over commutative Noetherian rings. In this case representability is understood in a more general sense and finitely generated relatively free algebras are embeddable in algebras which are finite-dimensional over their centers. Hence representability may be not realized by matrices. As a consequence, the author obtains that finitely generated relatively free algebras over Noetherian commutative rings are residually finite. The main results were announced in the proceedings and the abstracts of several conferences in 1998–2000. The paper also contains interesting discussions, with related problems and a comprehensive list of references. Reviewed by Vesselin Drensky

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