Definition:Norm/Division Ring

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

A (multiplicative) norm on $R$ is a mapping from $R$ to the non-negative reals:

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:Field Norm of Quaternion, which is actually not a norm as shown in Field Norm of Quaternion is not Norm.
 * Definition:Norm on Vector Space


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