Definition:Norm/Ring
This page is about Norm on Ring. For other uses, see 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:
| \((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall x \in R:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = 0_R \) | |||
| \((\text N 2)\) | $:$ | Submultiplicativity: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x \circ y} \) | \(\ds \le \) | \(\ds \norm x \times \norm y \) | |||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Normed Ring
Let $\norm {\, \cdot \,}$ be a norm on $R$.
Then $\struct {R, \norm {\, \cdot \,} }$ is a normed ring.
Notes
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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 division 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
- Definition:Absolute Value, a well known norm as shown in Absolute Value is Norm.
- Definition:Complex Modulus, a well known norm as shown in Complex Modulus is Norm.
- Definition:Field Norm of Quaternion, which is actually not a norm, as shown in Field Norm of Quaternion is not Norm.