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


Authors: Alexei Belov-Kanel, Louis Rowen and Uzi Vishne
Journal: Trans. Amer. Math. Soc. 367 (2015), 5553-5596
MSC (2010): Primary 16R10; Secondary 16G20
Published electronically: April 3, 2015
MathSciNet review: 3347183
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Abstract: In a series of papers by the authors we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a ``$ \bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $ \Gamma $'', i.e., a nonidentity of the representable algebra $ A$ arising from $ \Gamma $, whose ideal of evaluations in $ A$ is closed under multiplication by $ \bar q$-powers of the characteristic coefficients of matrices corresponding to the generators of $ A$, where $ \bar q$ is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring $ C$ involves localizing at finitely many elements a kind of $ C$, and reducing to the field case by a local-global principle.


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Additional Information

Alexei Belov-Kanel
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: belova@math.biu.ac.il

Louis Rowen
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: rowen@math.biu.ac.il

Uzi Vishne
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: vishne@math.biu.ac.il

DOI: https://doi.org/10.1090/tran/5983
Received by editor(s): September 2, 2012
Received by editor(s) in revised form: June 13, 2013
Published electronically: April 3, 2015
Additional Notes: This work was supported by the Israel Science Foundation (grant no. 1207/12).
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.