Norm on a commutative unital ring: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>R</math> be a [[commutative unital ring]]. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the | Let <math>R</math> be a [[commutative unital ring]]. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the integers. | ||
The term '''norm''' is typically used for a [[nonnegative norm]]: a norm that always takes nonnegative values. | |||
The [[algebraic norm in a number field]], restricted to its [[ring of integers in a number field|ring of integers]], is ''not'' a nonnegative norm in this sense for all number fields because it is not always necessarily nonnegative. | |||
Norms are typically used for integral domains. | Norms are typically used for integral domains. | ||
Latest revision as of 02:26, 24 January 2009
Definition
Let be a commutative unital ring. A norm on is a function from the nonzero elements of to the integers.
The term norm is typically used for a nonnegative norm: a norm that always takes nonnegative values.
The algebraic norm in a number field, restricted to its ring of integers, is not a nonnegative norm in this sense for all number fields because it is not always necessarily nonnegative.
Norms are typically used for integral domains.
Properties
For a complete list of properties, refer:
Category:Properties of norms on commutative unital rings
Multiplicative norm
Further information: Multiplicative norm
A norm on a commutative unital ring is termed multiplicative if the norm of a nonzero product of two elements is the product of their norms.
Characteristic norm
Further information: Characteristic norm
A norm on a commutative unital ring is termed characteristic if it is invariant under all automorphisms of the ring.
Other typical norm properties
- Multiplicatively monotone norm: The norm of a product is at least equal to the norm of each of the factors.
- Filtrative norm: The set of elements of norm less than , along with zero, forms an additive subgroup for any .