Specht’s problem for associative affine algebras over commutative Noetherian rings
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- by Alexei Belov-Kanel, Louis Rowen and Uzi Vishne PDF
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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.References
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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
- MR Author ID: 151270
- Email: rowen@math.biu.ac.il
- Uzi Vishne
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
- MR Author ID: 626198
- ORCID: 0000-0003-2760-9775
- Email: vishne@math.biu.ac.il
- 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).
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 5553-5596
- MSC (2010): Primary 16R10; Secondary 16G20
- DOI: https://doi.org/10.1090/tran/5983
- MathSciNet review: 3347183