Definition:Norm/Ring

This page is about the 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 (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 $

Let $\norm{\,\cdot\,}$ be a norm on $R$.

Then $\struct{R, \norm{\,\cdot\,} }$ is a normed ring.

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

\((N1)\)	$:$	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:	\(\displaystyle \forall x, y \in R:\)	\(\displaystyle \norm {x + y} \)	\(\displaystyle \le \)	\(\displaystyle \norm x + \norm y \)