Definition:Norm on Division Ring

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

A norm on $R$ is a map on $R$ to the nonnegative reals $\left\Vert{\cdot}\right\Vert: R \to \R_+ \cup \left\{{0}\right\}$ satisfying the following three properties (for all $x,y \in R$):

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

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

Also, a field $k$ that is endowed with a norm is known as a valued field.

Also See

 * Definition:Modulus of Complex Number, a well-known example of a norm.
 * Definition:Norm (Vector Space), an extension to vector spaces.