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

TL;DR

This paper investigates NIP rings, especially $F_p$-algebras, proving they decompose into Henselian local rings and establishing properties like prime ideal definability and Artin-Schreier surjectivity.

Contribution

It establishes that NIP $F_p$-algebras are finite products of Henselian local rings and proves surjectivity of the Artin-Schreier map in this context.

Findings

Prime and radical ideals are externally definable in NIP rings.

Localization of NIP rings remains NIP.

$F_p$-algebras are finite products of Henselian local rings.

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).