Valuation Ring - Examples

Examples

• Any field is a valuation ring.

• Z_(p), the localization of the integers Z at the prime ideal (p), consisting of ratios where the numerator is any integer and the denominator is not divisible by p. The field of fractions is the field of rational numbers Q.

• The ring of meromorphic functions on the entire complex plane which have a Maclaurin series (Taylor series expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If f does not have a Maclaurin series then 1/f does.

• Any ring of p-adic integers Z_p for a given prime p is a local ring, with field of fractions the p-adic numbers Q_p. The algebraic closure Z_pcl of the p-adic integers is also a local ring, with field of fractions Q_pcl. Both Z_p and Z_pcl are valuation rings.

• Let k be an ordered field. An element of k is called finite if it lies between two integers n<x<m; otherwise it is called infinite. The set D of finite elements of k is a valuation ring. The set of elements x such that x ∈ D and x−1∉D is the set of infinitesimal elements; and an element x such that x∉D and x−1∈D is called infinite.

• The ring F of finite elements of a hyperreal field *R is a valuation ring of *R. F consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, which is equivalent to saying a hyperreal number x such that −n < x < n for some standard integer n. The residue field, finite hyperreal numbers modulo the ideal of infinitesimal hyperreal numbers, is isomorphic to the real numbers.