Polynomial-mappings and M-equivalence

Let R be an integral domain which is either finitely generated over its prime subring or a Noetherian domain with only finite residue fields and only finitely many units. Let f be a univariate polynomial of degree ⩾2 having coefficients in R and let E be an infinite subset of R. Then, we prove the existence of a maximal ideal m of R such that E and f(E) have distinct m-adic closures. As a corollary, we derive some results on polynomial equivalence and full-invariance of subsets under polynomial mappings.