6 - Rings of definable scalars

Summary

Rings of pp-definable scalars, and the more general pp-type-definable scalars, are defined and their basic properties developed. Section 12.8 gives another way of arriving at these rings.

Rings of definable scalars

Rings of definable scalars are defined in Section 6.1.1. In Section 6.1.2 it is shown that every element of an epimorphic extension of a ring is definable (in its actions on modules) over that ring. An example is given to show that the notion of localisation implicit in rings of definable scalars does not always yield an epimorphism of rings.

Classical localisations are shown in Section 6.1.3 to be examples of rings of definable scalars. Duality preserves rings of definable scalars (Section 6.1.4). The rings of definable scalars of the points of the spectrum of a PI Dedekind domain are computed in Section 6.1.5.

In Section 6.1.6 we allow scalars defined by pp-types, that is, by infinite sets of pp conditions. These are compared with rings of definable scalars and used to show that rings of definable scalars can be realised as biendomorphism rings.

Actions defined by pp conditions

Scalars defined by pp conditions are those which extend across definable subcategories (6.1.1). If a closed subset contains the support of RR, then its ring of definable scalars is R (6.1.5). The ring of definable scalars is just a part of the category of pp-pairs (6.1.7).