Fields and rings with few types

TL;DR

This paper explores the properties of weakly small rings and fields, revealing structural constraints and classifications, such as finiteness or algebraic closure, and characterizing division rings and radicals.

Contribution

It provides new classification results for weakly small fields and rings, including their extensions, radicals, and division ring structures, under various conditions.

Findings

Weakly small fields of characteristic 2 are finite or algebraically closed.

Every weakly small division ring of positive characteristic is locally finite dimensional over its center.

The Jacobson radical of a weakly small ring is locally nilpotent.

Abstract

Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It is shown here that a field extension of finite degree of a weakly small field either is a finite field or has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. Every weakly small division ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small ring is locally nilpotent. Every weakly small division ring is locally, modulo its Jacobson radical, isomorphic to a product of finitely many matrix rings over division rings.