Let $D$ be a domain which is not a field. If there exists $b \in D - \{0\}$ such that for all $a \in D - \{0\}$, $a|b$, then is it true that $b = 0$?
This is certainly true if one has a UFD with infinitely many irreducibles (that are not associates) or a Jacobson domain which is not a field (or more generally a domain where the intersection of all non-zero prime ideals is the zero ideal), but I can't seem to be able to prove this for a domain in general or find a counterexample.
If this statement is false for an arbitrary domain, then is there some additional weak hypothesis on the domain which makes the above statement true?
I was actually trying to answer a question asked yesterday on MSE, which was the following:
If $D$ is a domain which is not a field and $Q = Frac(D)$, then $Hom_D(Q,D) = \{0\}$. Note that if $\varphi$ is any such $D$-linear map, then for all $a \in D- \{0\}$, $a\varphi(1/a) = \varphi(a/a) = \varphi(1)$. Thus, I get for all $a \in D - \{0\}$, $a|\varphi(1)$, and I want to conclude that $\varphi(1) = 0$, but cannot.