Principal ideal domain

Definition

A principal ideal domain is an integral domain $R$ in which every ideal is principal. That is, for every ideal $I$ of $R$, there is $a \in R$ with $I = (a)$.

Remarks

Every Euclidean domain is a principal ideal domain. So $\mathbb{Z}$, $K[x]$ for a field $K$, and $\mathbb{Z}[i]$ are principal ideal domains. The ring $\mathbb{Z}[x]$ is not a principal ideal domain. Every principal ideal domain is a unique factorization domain. The ring $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.

Explanations

L1 human

In a principal ideal domain, every ideal is the set of the multiples of one element. For the integers, every ideal is $n\mathbb{Z}$ for some $n$. This makes divisibility in a principal ideal domain behave much like divisibility of integers.