In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element).
Some authors such as Bourbaki refer to PIDs as principal rings.
Principal ideal domains are mathematical objects that behave like the integers, with respect to divisibility: any element of a PID has a unique factorization into prime elements (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 x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by, etc.
Principal ideal domains appear in the following chain of class inclusions: Examples include: Examples of integral domains that are not PIDs: The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then
[5] This does not hold for modules over arbitrary rings, as the example
In a principal ideal domain, any two elements a,b have a greatest common divisor, which may be obtained as a generator of the ideal (a, b).
,[6][7] this was proved by Theodore Motzkin and was the first case known.
[9][10][11][12] The converse does not hold since for any UFD K, the ring K[X, Y] of polynomials in 2 variables is a UFD but is not a PID.
It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element.)
The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.
Let A be an integral domain, the following are equivalent.
Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID.
(4) compares to: An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.
A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.