TITLE: Non-UFD integral domain such that prime is equivalent to irreducible?
QUESTION [11 upvotes]: In the integral domain every prime is irreducible. But the converse is not true, for example, $1+\sqrt{-3}$ is an irreducible but not a prime in ${\Bbb Z}[\sqrt{-3}]$. In a UFD, "prime" and "irreducible" are equivalent. 
Here is my question:

Is there a non-UFD integral domain such that prime is equivalent to irreducible?

REPLY [12 votes]: If $R$ is a GCD domain, then prime is equivalent to irreducible. Since any Bezout domain is a GCD domain, it is enough to find a Bezout domain which is not an UFD. Such examples can be found here. 
A classical example is $R=\mathbb Z+X\mathbb Q[X]$. $R$ is not an UFD since we have the following (strictly) ascending chain of principal ideals: $(X)\subset (\frac{1}{2}X)\subset\cdots\subset(\frac{1}{2^n}X)\subset\cdots$. Note that $R$ is not a "trivial" example, that is, $R$ has irreducible elements, since any prime number is irreducible in $R$. I leave you as an exercise to prove that $R$ is a GCD domain.