Nilpotent polynomials with non-nilpotent coefficients

Abstract: It is well known that the coefficients of nilpotent polynomials over noncommutative rings generally are not all nilpotent. We show that this remains true even under extremely strong restrictions on the set of nilpotents in the coefficient ring. If $R$ is a ring and its set of nilpotents, ${\rm Nil}(R)$, satisfies ${\rm Nil}(R)^2=0$, then in general ${\rm Nil}(R[x])\not \subseteq {\rm Nil}(R)[x]$. This is proven by constructing an explicit polynomial example. The smallest possible degree of such a polynomial is seven. Related problems are raised, as well as connections to Kothe's conjecture and work of Agata Smoktunowicz.

rings and algebrasrepresentation theory

Audience: researchers in the topic

ONCAS Online Noncommutative Algebra Seminar

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