# Definition:Norm/Ring

*This page is about the norm on a ring. For other uses, see Definition:Norm.*

## Contents

## 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**:

\((N1)\) | $:$ | Positive Definiteness: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||

\((N2)\) | $:$ | Submultiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x \times \norm y \) | ||

\((N3)\) | $:$ | Triangle Inequality: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |

### 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 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.