# 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.

## 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 $1.5$