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).

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