Submission date: 3 November 2021

Abstract:

We prove an assortment of results on (commutative and unital) NIP rings, especially 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 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/ℳ is infinite. Then the Artin-Schreier map R → R is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for fields).

Mathematics Subject Classification: 03C60

Keywords and phrases:

Full text arXiv 2111.02095: pdf, ps.