Definition:Norm/Ring

(Redirected from Definition:Norm on Ring)

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:

 $(\text N 1)$ $:$ Positive Definiteness: $\displaystyle \forall x \in R:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = 0_R$ $(\text N 2)$ $:$ Submultiplicativity: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x \circ y}$ $\displaystyle \le$ $\displaystyle \norm x \times \norm y$ $(\text N 3)$ $:$ 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.