Model theory of finite and pseudofinite rings

The model theory of finite and pseudofinite fields as well as the model theory of finite and
pseudofinite groups have been and are thoroughly studied. A close relation has been found
between algebraic and model theoretic properties of pseudofinite fields and psedudofinite
groups.
In this thesis we present results contributing to the beginning of the study of model
theory of finite and pseudofinite rings.
In particular we classify the theory of ultraproducts of finite residue rings in the context
of generalised stability theory. We give sufficient and necessary conditions for the theory
of such ultraproducts to be NIP, simple, NTP2 but not simple nor NIP, or TP2 .
Further, we show that for any fixed positive l in N the class of finite residue rings
{Zp=p^l Zp : p in P} forms an l-dimensional asymptotic class. We discuss related classes
of finite residue rings in the context of R-multidimensional asymptotic classes.
Finally we present a classification of simple and semisimple (in the algebraic sense)
pseudofinite rings, we study NTP2 classes of J-semisimple rings and we discuss NIP
classes of finite rings and ultraproducts of these NIP classes.