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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ G$-graded central polynomials and $ G$-graded Posner's theorem


Author: Yakov Karasik
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16R20; Secondary 16R10, 16R50
DOI: https://doi.org/10.1090/tran/7736
Published electronically: January 4, 2019
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Abstract: Let $ \mathbb{F}$ be a characteristic zero field, let $ G$ be a residually finite group, and let $ W$ be a $ G$-prime and polynomial identity $ \mathbb{F}$-algebra. By constructing $ G$-graded central polynomials for $ W$, we prove the $ G$-graded version of Posner's theorem. More precisely, if $ S$ denotes all nonzero degree $ e$ central elements of $ W$, the algebra $ S^{-1}W$ is $ G$-graded simple and finite dimensional over its center.

Furthermore, we show how to use this theorem in order to recapture a result of Aljadeff and Haile stating that two $ G$-simple algebras of finite dimension are isomorphix if and only if their ideals of graded identities coincide.


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Additional Information

Yakov Karasik
Affiliation: Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel
Email: theyakov@gmail.com

DOI: https://doi.org/10.1090/tran/7736
Received by editor(s): February 26, 2017
Received by editor(s) in revised form: October 7, 2018
Published electronically: January 4, 2019
Article copyright: © Copyright 2019 American Mathematical Society