$G$-graded central polynomials and $G$-graded Posner’s theorem
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- by Yakov Karasik PDF
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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
- MR Author ID: 1014910
- Email: theyakov@gmail.com
- Received by editor(s): February 26, 2017
- Received by editor(s) in revised form: October 7, 2018
- Published electronically: January 4, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 5531-5546
- MSC (2010): Primary 16R20; Secondary 16R10, 16R50
- DOI: https://doi.org/10.1090/tran/7736
- MathSciNet review: 4014286