[summary: A principal ideal domain is an Integral domain in which every [ideal_ring_theory ideal] has a single generator.]

In ring theory, an Integral domain is a **principal ideal domain** (or **PID**) if every [ideal_ring_theory 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_homomorphism 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 Euclidean domain] is a PID. (Proof.)
- Therefore $~$\mathbb{Z}$~$ is a PID, because it is a [euclidean_domain 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 [polynomial_ring ring $~$F[X]$~$ of polynomials] over a field $~$F$~$ is a PID, because it is a Euclidean domain. (Its Euclidean function is "take the [polynomial_degree degree] of the polynomial".)
- The ring of [gaussian_integer Gaussian integers], $~$\mathbb{Z}[i]$~$, is a PID because it is a Euclidean domain. ([gaussian_integers_is_pid Proof]; its Euclidean function is "take the [norm_complex_number 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. [todo: 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. ([principal_ideal_domain_has_unique_factorisation 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_ideal maximal ideals] of PIDs.
- Every PID is trivially [noetherian_ring Noetherian]: every ideal is not just
*finitely*generated, but generated by a single element.