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Nilpotent polynomials and nilpotent coefficients
Draper, Thomas L. ; Nielsen, Pace P. ; Šter, Janez, matematik
For any ring ▫$R$▫, let ▫${\rm Nil}(R)$▫ denote the set of nilpotent elements in ▫$R$▫, and for any subset ▫$S \subseteq R$▫, 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 ▫${\rm Nil}(R[x])$▫ is a proper subset of ▫${\rm Nil}(R)[x]$▫. In this paper we give an example in the converse direction: there exists a ring ▫$R$▫ such that ▫${\rm Nil}(R)[x]$▫ is a proper subset of ▫${\rm Nil}(R[x])$▫. This is achieved by constructing a ring ▫$R$▫ with ▫${\rm Nil}(R)^2 = 0$▫ and a polynomial ▫$f \in R[x] \setminus {\rm Nil}(R)[x]$▫ satisfying ▫$f^2 = 0$▫. The smallest possible degree of such a polynomial is seven. The example we construct answers an open question of Antoine related to Armendariz rings.
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