#### Vol. 1, No. 1, 2022

 Recent Issues Volume 1, Issue 1
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Henselianity in NIP $\mathbb{F}_p$-algebras

### Will Johnson

Vol. 1 (2022), No. 1, 115–128
DOI: 10.2140/mt.2022.1.115
##### 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).

##### Keywords
NIP, henselian rings
Primary: 03C60