Principal ideal domain

by Patrick Stevens Aug 3 2016 updated Aug 4 2016

A principal ideal domain is a kind of ring, in which all ideals have a certain nice form.

[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".


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.)