Everywhere vanishing polynomial functions

摘要

If R is a finite commutative ring, it is well known that there exists a nonzero polynomial in R[T] which is satisfied by every element of R. In this paper, we classify all commutative rings R such that every element of R satisfies a particular monic polynomial. If the polynomial, satisfied by the elements of R, is not required to be monic, then we can give a classification only for Noetherian rings, giving examples to show that the characterization does not extend to arbitrary commutative rings.