I was trying to get a feel for what it means for an element to be irreducible but not prime, and I came across the idea that an irreducible nonprime element $r$ has "hidden" factors. That is, the reason $r|ab$ but $r\not|a$ and $r\not | b$ is that we should be able to write $r = p q$, but $p$ and $q$ are missing from the ring. Using the canonical example of $\mathbb Z[x^2,x^3]\subset \mathbb Z[x]$, the polynomials with integer coefficients but no linear term, $x^2$ and $x^3$ are irreducible but not prime. Here the "hidden factor" is obvious: $x$, and the closure of $\mathbb Z[x^2,x^3]\cup \{x\}$ in $\mathbb Z[x]$ is $\mathbb Z[x]$ itself, where every irreducible element is prime.
So my question is whether this idea of "hidden" factors holds in general. Let $R$ be an integral domain and let $I(R)$ be the set of all irreducible nonprime elements of $R$. Does there exist another integral domain $P$ such that $R$ is a subring of $P$, $I(P) = \emptyset$, and the closure of $R \cup \{p\in P : \exists r\in I(R)( p|r)\}$ in $P$ is $P$ itself?
I'm also curious if there is a unique minimal $P$, as well as if $P$ can be chosen to have stronger properties than $I(P) = 0$, such as being a UFD.