# Definition:Norm/Ring/Multiplicative

< Definition:Norm | Ring

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

## Definition

Let $\struct {R, +, \circ}$ be a 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: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |