Editing Principal ideal (section)

==Properties==
Any [[Euclidean domain]] is a [[Principal ideal domain|PID]]; the algorithm used to calculate [[greatest common divisor]]s may be used to find a generator of any ideal.
More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication.
In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a [[unit (ring theory)|unit]]; we define <math>\gcd(a, b)</math> to be any generator of the ideal <math>\langle a, b \rangle.</math>

For a [[Dedekind domain]] <math>R,</math> we may also ask, given a non-principal ideal <math>I</math> of <math>R,</math> whether there is some extension <math>S</math> of <math>R</math> such that the ideal of <math>S</math> generated by <math>I</math> is principal (said more loosely, <math>I</math> ''becomes principal'' in <math>S</math>).
This question arose in connection with the study of rings of [[algebraic integer]]s (which are examples of Dedekind domains) in [[number theory]], and led to the development of [[class field theory]] by [[Teiji Takagi]], [[Emil Artin]], [[David Hilbert]], and many others.

The [[principal ideal theorem|principal ideal theorem of class field theory]] states that every integer ring <math>R</math> (i.e. the [[ring of integers]] of some [[number field]]) is contained in a larger integer ring <math>S</math> which has the property that ''every'' ideal of <math>R</math> becomes a principal ideal of <math>S.</math> In this theorem we may take <math>S</math> to be the ring of integers of the [[Hilbert class field]] of <math>R</math>; that is, the maximal [[Ramification (mathematics)|unramified]] abelian extension (that is, [[Galois extension]] whose [[Galois group]] is [[abelian group|abelian]]) of the fraction field of <math>R,</math> and this is uniquely determined by <math>R.</math>

[[Krull's principal ideal theorem]] states that if <math>R</math> is a Noetherian ring and <math>I</math> is a principal, proper ideal of <math>R,</math> then <math>I</math> has [[height (ring theory)|height]] at most one.