Values of polynomials on centrally closed prime algebras

Abstract

Let [Formula: see text] be a simple algebra over its extended centroid [Formula: see text], and let [Formula: see text] be a noncommutative polynomial having zero constant term. We denote by [Formula: see text] the additive subgroup of [Formula: see text] generated by all elements [Formula: see text] for [Formula: see text]. It is proved that if [Formula: see text], then [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text]. As to the case [Formula: see text], an example of a polynomial [Formula: see text] satisfying [Formula: see text] is given. Also, the polynomials [Formula: see text] with [Formula: see text] are characterized if [Formula: see text] and [Formula: see text]. Moreover, we work on the context of centrally closed prime algebras to get more general results.