ground ring

A **ground ring** (or **base ring**, **field of scalars**, etc) is a ring $k$ (usually a commutative ring or even a field) which is fixed in a given situation, such that everything takes place ‘over’ $k$. 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 $k$, although its meaning does depend on $k$.

There are analogies between the ground ring and the base space of a bundle. There are also generalisations in which $k$ 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.

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 $k$-module (either left or right when $k$ might not be commutative), and a vector space is not just a vector space but a vector space over $k$. 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:

- graded module
- graded vector space
- algebraic variety
- relative scheme
- 2-vector space
- category algebra
- groupoidification
- Hopf algebra
- symmetric function
- coalgebra
- associative algebra
- nonassociative algebra
- fusion category
- vector bundle
- almost scheme
- tangent vector
- cotangent vector
- exterior algebra
- Hilbert space
- Banach space
- topological vector space
- Legendre polynomial
- differential form
- inner product space
- affine space

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

Revised on January 10, 2013 02:51:01
by Toby Bartels
(64.89.53.241)