TITLE: Is there an integral domain such that none of its irreducible elements is prime?
QUESTION [11 upvotes]: Is there some integral domain such that none of its irreducible elements is prime? 

Recall that a nonzero, non invertible element $a$ of an integral domain $D$ is said to be

Irreducible, if for all $b,c\in D$ such that $a=bc$, then either $b$ or $c$ is a unit in $D$.
Prime, if for all $b,c\in D$ such that $a$ divides $bc$, then either $a$ divides $b$ or $a$ divides $c$.

Clearly prime implies irreducible. The converse is not true in general, but is valid when $D$ is a UFD. Obviously these notions are vacuously equivalent if $D$ has not irreducible elements (see here for examples).
Summarizing, I want to know if there is some integral domain with at least one irreducible element, but without prime elements.

REPLY [4 votes]: Hint $ $ The irreducibles of $\,\Bbb Q+x\Bbb R[[x]]\,$ are $\,rx\ne 0,\,r\in \Bbb R.\,$ But $\,rx\mid (\pi rx)^2,\,\ rx\nmid \pi r x\,$ by $\,\pi\not\in\Bbb Q$