Definition:Norm/Ring

This page is about Norm on Ring. For other uses, see Norm.

Definition

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

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

$\norm {\,\cdot\,}: R \to \R_{\ge 0}$

satisfying the (ring) submultiplicative norm axioms:

\((\text N 1)\)	$:$	Positive Definiteness:	\(\displaystyle \forall x \in R:\)		\(\displaystyle \norm x = 0 \)	\(\displaystyle \iff \)	\(\displaystyle x = 0_R \)
\((\text N 2)\)	$:$	Submultiplicativity:	\(\displaystyle \forall x, y \in R:\)		\(\displaystyle \norm {x \circ y} \)	\(\displaystyle \le \)	\(\displaystyle \norm x \times \norm y \)
\((\text N 3)\)	$:$	Triangle Inequality:	\(\displaystyle \forall x, y \in R:\)		\(\displaystyle \norm {x + y} \)	\(\displaystyle \le \)	\(\displaystyle \norm x + \norm y \)

Normed Ring

Let $\norm {\, \cdot \,}$ be a norm on $R$.

Then $\struct {R, \norm {\, \cdot \,} }$ is a normed ring.

Notes

In contrast to the definition of a norm on a ring, a division ring norm is always assumed to be a multiplicative norm.

The reason for this is by Normed Vector Space Requires Multiplicative Norm on Division Ring, the norm on a division ring that is the scalar division ring of a normed vector space must be a multiplicative norm.

By Ring with Multiplicative Norm has No Proper Zero Divisors it follows that a ring with zero divisors has no multiplicative norms, so a multiplicative norm is too restrictive for a general ring.

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