Definition:Norm/Division Ring

Definition
Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

A norm on $R$ is a mapping from $R$ to the non-negative reals:
 * $\norm {\,\cdot\,}: R \to \R_{\ge 0}$

satisfying the (division ring) norm axioms:

If $\norm {\,\cdot\,}$ satisfies the weaker axiom:

then $\norm {\,\cdot\,}$ is called a submultiplicative norm on $R$.

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 common to define the norm only for subfields of the complex numbers.

However, the definition given here incorporates this approach.

Also see

 * Definition:Absolute Value, a well known norm as shown in Absolute Value is Norm.
 * Definition:Complex Modulus, a well known norm as shown in Complex Modulus is Norm.
 * Definition:Norm of Quaternion, a well known norm as shown in Quaternion Norm is Norm.
 * Definition:Norm on Ring
 * Definition:Norm on Vector Space


 * Definition:Normed Division Ring
 * Definition:Topological Ring
 * Definition:Valued Field