Polynomials That Force a Unital Ring to be Commutative

We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f ( R ) = 0, in the sense that f ( x ) = 0 for all . Such a polynomial must be primitive, and for primitive polynomials the condition f ( R ) = 0 forces R to have nonzero characteristi...

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Bibliographic Details
Published in:Resultate der Mathematik 2013-09, Vol.64 (1-2), p.59-65
Main Authors: Buckley, S. M., MacHale, D.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f ( R ) = 0, in the sense that f ( x ) = 0 for all . Such a polynomial must be primitive, and for primitive polynomials the condition f ( R ) = 0 forces R to have nonzero characteristic. The task is then reduced to considering rings of prime power characteristic and the main step towards the full characterization is a characterization of polynomials f such that R is necessarily commutative if f ( R ) = 0 and R is a unital ring of characteristic some power of a fixed prime p .
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-012-0296-0