Definition:Norm/Ring

Definition
Let $\struct {R, +, \circ}$ be a 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 (ring) norm axioms:

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

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

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 Division Ring
 * Definition:Norm on Vector Space
 * Definition:Topological Ring