Definition:Norm/Ring/Multiplicative

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This page is about Multiplicative Norm on Ring. For other uses, see 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:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\ds \forall x \in R:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = 0_R \)      
\((\text N 2)\)   $:$   Multiplicativity:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x \circ y} \)   \(\ds = \)   \(\ds \norm x \times \norm y \)      
\((\text N 3)\)   $:$   Triangle Inequality:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \norm x + \norm y \)      


Also see