Abstract
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 \({x \in R}\). 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.
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Buckley, S.M., MacHale, D. Polynomials That Force a Unital Ring to be Commutative. Results. Math. 64, 59–65 (2013). https://doi.org/10.1007/s00025-012-0296-0
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DOI: https://doi.org/10.1007/s00025-012-0296-0