nLab
ground ring

Contents

Idea

A ground ring (or base ring, field of scalars, etc) is a ring kk (usually a commutative ring or even a field) which is fixed in a given situation, such that everything takes place ‘over’ kk. There is no technical definition here; rather, it is the meaning of ‘over’ that must be defined in any particular case.

The elements of the ground ring are often called scalars. Note that a ‘scalar field’ in the sense of physics does not refer to kk, although its meaning does depend on kk.

There are analogies between the ground ring and the base space of a bundle. There are also generalisations in which kk might be, for example, a monad. It is also important to consider base change from one ground ring to another, mediated by a ring homomorphism or even a bimodule.

Examples

Perhaps most fundamentally, the categories Mod and Vect depend on (respectively) a ground ring and a ground field. That is, a module is not just a module but a kk-module (either left or right when kk might not be commutative), and a vector space is not just a vector space but a vector space over kk. Conversely, the ground ring/field itself appears as the unit of the tensor product in these categories.

Other terms which depend on a ground ring/field include:

This list is very incomplete, made mostly by searching for ‘ground ring’, ‘base field’, etc.

Revised on November 11, 2014 12:39:58 by Urs Schreiber (89.204.130.78)