Example of an integral domain with a non-principal prime ideal of height one [closed]

Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of height $1$ which is not a principal ideal?

• What have you tried? Are you familiar with Dedekind domains which are not principal ideal domains? – Mohan Feb 5 '16 at 21:21

It is proved here that the ideal $\mathfrak p=(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$ is not principal. This is a maximal ideal of height one.
• I am confused: $(0)$ a prime ideal in this ring since it is a domain. There is a chain $(0)\subsetneq (2)\subsetneq \mathfrak{p}$ of ideals. Your answer implies that $(2)$ is not a prime ideal which means that $2$ is not a prime element in this ring. Is this true? – user8463524 Feb 5 '16 at 21:58