Principal ideal domain
Call the ring we are working in $A$.
Definition. A principal ideal domain is an integral domain in which every ideal is principal, meaning it is generated by a single element.
Properties
- Every non-zero prime ideal is maximal.
Proofs
1
Let $(p)$ be a a non-zero prime ideal of $A$. Suppose $I$ is another ideal that strictly contains it. We are in a PID, so it is generated by a single elements; write $I=(x)$. So $p\in (x)$ so let $p=xy$. But then $xy\in (p)$ when $x\notin (p)$ by construction. Since $(p)$ is prime, it must be that $y\in (p)$. So let $y=tp$. But then $p=xy=xtp$ thus $xt=1$ thus $1=xt\in (x)$ so $(x)=(1)$.