Polynomials That Force a Unital Ring to be Commutative |
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Authors: | S. M. Buckley D. MacHale |
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Affiliation: | 1. Department of Mathematics and Statistics, National University of Ireland Maynooth, Maynooth, Co., Kildare, Ireland 2. School of Mathematical Sciences, University College Cork, Cork, Ireland
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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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