In mathematics, a principalideal 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 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 ; any two elements of a PID have a greatest common divisor. If and are elements of a PID without common divisors, then every element of the PID can be written in the form. 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:
is an example of a ring which is not a unique factorization domain, since Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains.
: the ring of all polynomials with integer coefficients. It is not principal because is an example of an ideal that cannot be generated by a single polynomial.
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 principle ideal domains. 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.
Modules
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 . If M is a free module over a principal ideal domainR, then every submodule of M is again free. This does not hold for modules over arbitrary rings, as the example of modules over shows.
Properties
In a principal ideal domain, any two elements have a greatest common divisor, which may be obtained as a generator of the ideal. 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 and exist, with, so that, despite and having a greatest common divisor of. Every principal ideal domain is a unique factorization domain. The converse does not hold since for any UFD, the ring of polynomials in 2 variables is a UFD but is not a PID.
In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
All principal ideal domains are integrally closed.
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.
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 gives yet another proof that a PID is a UFD.
