nLab ground ring

Idea

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.

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 $k$-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.

Related concepts

• positive characteristic