Definition:Norm/Division Ring
This page is about Norm on Division Ring. For other uses, see Norm.
Definition
Let $\struct {R, +, \circ}$ be a division 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: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Notes
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In contrast to the definition of a norm on a division ring, a ring norm is always assumed to be a submultiplicative 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 known as
Some authors refer to this concept as an (abstract) absolute value on $R$.
A field that is endowed with a norm is thereby referred as a valued field.
Also defined as
In the literature, it is common to define the norm only for subfields of the complex numbers.
However, the definition given here incorporates this approach.
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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.
- Definition:Norm on Vector Space
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.1$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (next): $\S 1.2$: Normed Fields: Definition $1.5$