In ring theory, an integral domain is a principal ideal domain (or PID) if every ideal can be generated by a single element. That is, for every ideal \(I\) there is an element \(i \in I\) such that \(\langle i \rangle = I\); equivalently, every element of \(I\) is a multiple of \(i\).

Since ideals are kernels of ring homomorphisms (proof), this is saying that a PID \(R\) has the special property that every ring homomorphism from \(R\) acts “nearly non-trivially”, in that the collection of things it sends to the identity is just “one particular element, and everything that is forced by that, but nothing else”.

Examples

• Every Euclidean domain is a PID. (Proof.)

• Therefore \(\mathbb{Z}\) is a PID, because it is a Euclidean domain. (Its Euclidean function is “take the modulus”.)

• Every field is a PID because every ideal is either the singleton \(\{ 0 \}\) (i.e. generated by \(0\)) or else is the entire ring (i.e. generated by \(1\)).

• The ring \(F[X]\) of polynomials over a field \(F\) is a PID, because it is a Euclidean domain. (Its Euclidean function is “take the degree of the polynomial”.)

• The ring of Gaussian integers, \(\mathbb{Z}[i]\), is a PID because it is a Euclidean domain. (Proof; its Euclidean function is “take the norm”.)

• The ring \(\mathbb{Z}[X]\) (of integer-coefficient polynomials) is not a PID, because the ideal \(\langle 2, X \rangle\) is not principal. This is an example of a unique factorisation domain which is not a PID. proof of this

• The ring \(\mathbb{Z}_6\) is not a PID, because it is not an integral domain. (Indeed, \(3 \times 2 = 0\) in this ring.)

There are examples of PIDs which are not Euclidean domains, but they are mostly uninteresting. One such ring is \(\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]\). (Proof.)

Properties

• Every PID is a unique factorisation domain. (Proof; this fact is not trivial.) The converse is false; see the case \(\mathbb{Z}[X]\) above.

• In a PID, “prime” and “irreducible” coincide. (Proof.) This fact also characterises the maximal ideals of PIDs.

• Every PID is trivially Noetherian: every ideal is not just finitely generated, but generated by a single element.