Henselianity in NIP ${\mathbb{F}}_p$-algebras

Abstract

We prove an assortment of results on (commutative and unital) NIP rings, especially ${\mathbb{F}}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is NIP. Suppose $R$ is additionally an ${\mathbb{F}}_p$-algebra. Then $R$ is a finite product of Henselian local rings. Suppose in addition that $R$ is integral. Then $R$ is a Henselian local domain, whose prime ideals are linearly ordered by inclusion. Suppose in addition that the residue field $R/\mathfrak{m}$ is infinite. Then the Artin-Schreier map $R\to R$ is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for fields).

14 pages

References in corpus (1)

• A Conjectural Classification of Strongly Dependent Fields