Definition:Norm on Division Ring

Definition
Let $\left({R, +, \circ}\right)$ be a division ring, and denote its zero by $0_R$.

Let $\left\Vert{\cdot}\right\Vert: R \to \R_{\ge 0}$ be a mapping, where $\R_{\ge 0}$ denotes the non-negative real numbers.

Then $\left\Vert{\cdot}\right\Vert$ is a norm on $R$ iff it satisfies the following properties:

These may be referred to as the (division ring) norm axioms.

Also known as
Some authors refer to this concept as an (abstract) absolute value on $R$.

A field that is endowed with a norm is thereby referred as a valued field.

Also defined as
In the literature, it is more common to define the norm only for subfields of the complex numbers.

However, the definition given here incorporates this approach.

Also see

 * Definition:Complex Modulus, a well known norm as shown in Modulus is Norm.
 * Definition:Norm (Vector Space)