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Transactions of the American Mathematical Society

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

 

 

Nil and power-central polynomials in rings


Author: Uri Leron
Journal: Trans. Amer. Math. Soc. 202 (1975), 97-103
MSC: Primary 16A38
DOI: https://doi.org/10.1090/S0002-9947-1975-0354764-6
MathSciNet review: 0354764
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Abstract: A polynomial in noncommuting variables is vanishing, nil or central in a ring, $ R$, if its value under every substitution from $ R$ is 0, nilpotent or a central element of $ R$, respectively.

THEOREM. If $ R$ has no nonvanishing multilinear nil polynomials then neither has the matrix ring $ {R_n}$. THEOREM. Let $ R$ be a ring satisfying a polynomial identity modulo its nil radical $ N$, and let $ f$ be a multilinear polynomial. If $ f$ is nil in $ R$ then $ f$ is vanishing in $ R/N$. Applied to the polynomial $ xy - yx$, this establishes the validity of a conjecture of Herstein's, in the presence of polynomial identity. THEOREM. Let $ m$ be a positive integer and let $ F$ be a field containing no $ m$th roots of unity other than 1. If $ f$ is a multilinear polynomial such that for some $ n > 2{f^m}$ is central in $ {F_n}$, then $ f$ is central in $ {F_n}$.

This is related to the (non)existence of noncrossed products among $ {p^2}$-dimensional central division rings.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0354764-6
Keywords: Polynomial identities, nil polynomials, power-central polynomials, Herstein's conjecture, crossed products
Article copyright: © Copyright 1975 American Mathematical Society