Model theory of finite and pseudofinite rings

• Main Author: Bello Aguirre, Ricardo Isaac
• Other Authors: Macpherson, H. Dugald
• Published: University of Leeds 2016
• Subjects: 512
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.698817

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.