Definition:Norm/Division Ring

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

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:

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

satisfying the (ring) multiplicative norm axioms:

$(\text N 1)$	$:$	Positive Definiteness:	$\displaystyle \forall x \in R:$	$\displaystyle \norm x = 0 $	$\displaystyle \iff $	$\displaystyle x = 0_R $
$(\text N 2)$	$:$	Multiplicativity:	$\displaystyle \forall x, y \in R:$	$\displaystyle \norm {x \circ y} $	$\displaystyle = $	$\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 $

Notes

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

This article is complete as far as it goes, but it could do with expansion, in particular:
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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

Sources

1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.1$
2007: Svetlana Katok: p-adic Analysis Compared with Real ... (next): $\S 1.2$: Normed Fields: Definition $1.5$

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