We study ultraproducts of finite residue rings \ where \ is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter \ to determine if the resulting ultraproduct \ has simple, NIP, \ but not simple nor NIP, or \ theory, noting that all these four cases occur.

Motivated by the Ax–Kochen/Ershov principle, a large number of questions about Henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this article, we investigate the burden of Henselian valued fields in the three-sorted Denef–Pas language. If T is a theory of Henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of T is equal to the sum of the burdens of its value group and (...) residue field. As a consequence, T is NTP2 if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”. (shrink)

We prove the triviality of the Grothendieck ring of a $\mathbb{Z}$-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K$^2$ to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.