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 ``-characteristic coefficient-absorbing polynomial in each T-ideal '', i.e., a nonidentity of the representable algebra arising from , whose ideal of evaluations in is closed under multiplication by -powers of the characteristic coefficients of matrices corresponding to the generators of , where is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring involves localizing at finitely many elements a kind of , 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.