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.
Best Answer
I found this paper which in Example 2.2 (d) (page 4) claims that $K[[X^2,X^3]]$ is an integral domain with no prime elements but many irreducible elements, for any field $K$.