In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.
Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition 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.
Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains.
Principal ideal domains appear in the following chain of class inclusions:
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 is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to for some  (notice that may be equal to , in which case is ).
If M is a free module over a principal ideal domain R, then every submodule of M is again free. This does not hold for modules over arbitrary rings, as the example of modules over shows.
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).
All Euclidean domains are principal ideal domains, but the converse is not true. An example of a principal ideal domain that is not a Euclidean domain is the ring , this was proved by Theodore Motzkin and was the first case known. In this domain no q and r exist, with 0 ≤ |r| < 4, so that , despite and having a greatest common divisor of 2.
Every principal ideal domain is a unique factorization domain (UFD). 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. (To prove this look at the ideal generated by 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. Then 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.