Vol. 1, No. 1, 2022

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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 𝔽p-algebras. Let R be a NIP ring. Then every prime ideal or radical ideal of R is externally definable, and every localization S1R is NIP. Suppose R is additionally an 𝔽p-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 R𝔪 is infinite. Then the Artin–Schreier map R R is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for fields).

Keywords
NIP, henselian rings
Mathematical Subject Classification
Primary: 03C60
Milestones
Received: 4 November 2021
Revised: 7 December 2021
Accepted: 18 December 2021
Published: 24 June 2022
Authors
Will Johnson
School of Philosophy
Fudan University
Shanghai
China