Nilpotent polynomials and nilpotent coefficients

Abstract

For any ring R, let Nil(R) denote the set of nilpotent elements in R, and for any subset S⊆R, let S[x] denote the set of polynomials with coefficients in S. Due to a celebrated example of Smoktunowicz, there exists a ring R such that Nil(R[x]) is a proper subset of Nil(R)[x]. In this paper we give an example in the converse direction: there exists a ring R such that Nil(R)[x] is a proper subset of Nil(R[x]). This is achieved by constructing a ring R with Nil(R)2=0 and a polynomial f∈R[x]∖Nil(R)[x] satisfying f2=0. The smallest possible degree of such a polynomial is seven. The example we construct answers an open question of Antoine related to Armendariz rings.