Specht's problem for associative affine algebras over commutative Noetherian rings

Belov-Kanel, A.; Rowen, L.; Vishne, U.

Авторы: Belov-Kanel Alexei, Rowen Louis, Vishne Uzi
Журнал: Transactions of the American Mathematical Society
Том: 367
Год издания: 2015
Первая страница: 5553
Последняя страница: 5596
DOI: 10.1090/tran/5983
Аннотация: Let C be a commutative Noetherian ring. The well known Specht problem for associative algebras asks whether the ideal of identities of every associative C-algebra is finitely generated. Its positive solution in the case when C is a field of characteristic 0 (the problem as stated by Specht) was given by Kemer, and influenced heavily the theory and the methods since then. It is also well known that the answer to the Specht problem is negative when C is an infinite field of positive characteristic. The Specht problem can be stated in an equivalent form: Does every algebra over C satisfy the ACC on T-ideals? The answer to the problem is positive if one considers affine PI algebras over an infinite field; this result is due to Kemer. Later on the first named author of the paper under review obtained that the answer is also positive for affine algebras over a Noetherian base ring. The paper under review gives a full proof of the latter theorem. Let us mention that here the authors have managed to avoid some (or most) of the combinatorics used in the original proof of Belov. In fact the authors generalize the result in Theorem 7.6. Namely they show that the solution to the Specht problem is positive in a general setting. They prove that each T-ideal of the free algebra over the commutative Noetherian ring C is finitely generated as a T-ideal. We draw the readers’ attention to the following fact. One commonly defines a T-ideal I as an ideal in the free C-algebra which is closed under substitutions. Moreover one requires that the ideal (in C) of the coefficients of the elements of I coincides with C. The authors show in their Theorem 7.6 that the last requirement can be dropped and the answer to the Specht problem is still positive.Reviewer: Plamen Koshlukov (Campinas)
Добавил в систему: Белов Алексей Яковлевич

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