Definition:Norm/Ring

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


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