Norm on a commutative unital ring

Definition

Let $R$ be a commutative unital ring. A norm on $R$ is a function from the nonzero elements of $R$ 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:

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 $n$, along with zero, forms an additive subgroup for any $n$.