Editing Principal ideal domain (section)

{{Short description|Algebraic structure}}
{{for|a similar structure that is not necessarily commutative and may have zero-divisors|Principal ideal ring}}
In [[mathematics]], a '''principal ideal domain''', or '''PID''', is an [[integral domain]] (that is, a [[commutative ring]] without nonzero [[zero divisor]]s) in which every [[ideal (ring theory)|ideal]] is [[principal ideal|principal]] (that is, is formed by the multiples of a single element). Some authors such as [[Nicolas Bourbaki|Bourbaki]] refer to PIDs as '''principal rings'''.

Principal ideal domains are mathematical objects that behave like the [[integer]]s, with respect to [[Integral domain#Divisibility, prime and irreducible elements|divisibility]]: any element of a PID has a unique factorization into [[prime element]]s (so an analogue of the [[fundamental theorem of arithmetic]] holds); any two elements of a PID have a [[greatest common divisor]] (although it may not be possible to find it using the [[Euclidean algorithm]]). If {{math|''x''}} and {{math|''y''}} are elements of a PID without common divisors, then every element of the PID can be written in the form {{math|''ax'' + ''by''}}, etc.

Principal ideal domains are [[noetherian ring|Noetherian]], they are  [[integrally closed domain|integrally closed]], they are [[unique factorization domain]]s and [[Dedekind domain]]s.  All [[Euclidean domain]]s and all [[field (mathematics)|fields]] are principal ideal domains.

Principal ideal domains appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}
{{Algebraic structures |Ring}}