# Definition:Norm/Division Ring

This page is about the norm on a division ring or a field. For other uses, see Definition: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:

 $(N1)$ $:$ Positive Definiteness: $\displaystyle \forall x \in R:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = 0_R$ $(N2)$ $:$ Multiplicativity: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x \circ y}$ $\displaystyle =$ $\displaystyle \norm x \times \norm y$ $(N3)$ $:$ 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 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.