Rings of definable scalars and biendomorphism rings

Summary

Abstract

The definable additive endomorphisms of a module form a ring which we call the ring of definable scalars of the module. One is lead, by various routes – model theoretic and algebraic – to consider these endomorphisms and the rings they form. In this paper we show that these rings may be realised as biendomorphism rings of suitably saturated modules and also as endomorphism rings of certain functors. We also consider rings of type-definable scalars and the context of arbitrary sorts.

1. Introduction

2. Rings of definable scalars

3. Rings of type-definable scalars and biendomorphism rings

4. Scalars in arbitrary sorts and endomorphism rings of localised functors

Introduction

Let us consider a (right) module M over a ring R. The elements of R act as scalars on M but, on this particular module, other scalars may act. For instance on any torsionfree divisible ℤ-module the ring, ℚ, of rationals has an action extending the action of ℤ via the natural embedding of rings ℤ → ℚ. We require that such “scalars” commute with the R–endomorphisms of the module and hence that they should belong to the biendomorphism ring of the module. But we shall also require that our scalars be definable from the R–action, thus excluding some biendomorphisms.