We prove an assortment of results on (commutative and unital) NIP rings, especially
-algebras.
Let
be a NIP ring. Then every prime ideal or radical ideal of
is externally definable,
and every localization
is NIP. Suppose is
additionally an
-algebra.
Then
is a finite product of henselian local rings. Suppose in addition that
is integral.
Then
is a henselian local domain, whose prime ideals are linearly
ordered by inclusion. Suppose in addition that the residue field
is infinite. Then the
Artin–Schreier map
is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for
fields).