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.
: rings of polynomials in one variable with coefficients in a field. (The converse is also true, i.e. if is a PID then is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form ,
Most rings of algebraic integers are not principal ideal domains because they have ideals which are not generated by a single element. This is one of the main motivations behind Dedekind's definition of Dedekind domains since a prime integer can no longer be factored into elements, instead they are prime ideals. In fact many for the p-th root of unity are not principal ideal domains[clarification needed]. In fact, the class number of a ring of algebraic integers gives a notion of "how far away" it is from being a principal ideal domain.
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  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.)
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 UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals.
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.