Let $\mathcal{D} \,' = \mathcal{D} \,'(\mathbb{R})$ be the space of distributions (continuous linear functionals on $C_c^\infty (\mathbb{R})$). Given $f \in C^\infty(\mathbb{R})$, do we have an explicit description of the set $A_f = \{F \in \mathcal{D} \,' | fF = 0 \}$?
For example, the $\delta$ distribution has $x \delta(x) = 0$, an equality of distributions, so $\delta \in A_x$, and clearly any scaled version of $\delta$ is also in $A_x$. But are there other distributions (linearly) independent of $\delta$ also in $A_x$? Writing it out, we're looking for distributions $F \in \mathcal{D} \, '$ such that $\forall \phi \in C_c^\infty(\mathbb{R}), \; \langle xF(x),\phi(x) \rangle := \langle F(x) ,x \phi(x) \rangle = 0$. This prescribes the values of $F$ on the proper ideal generated by $x$, but apart from that, $F$ is unspecified. So, it seems reasonable to believe there be more distributions in $A_x$ that are (linearly) independent of $\delta$.
This question becomes relevant when "explicitly solving" certain ODEs for distributions (i.e. we look for solutions that are distributions, not just classical functions), and we end up trying to "divide" by a smooth function, e.g. $x$, along the way. To ensure that we're not losing any potential solutions, we need to keep track of those distributions $F \in A_x$.
Thinking of $\mathcal{D} \,'$ as a module over the ring of smooth functions, $C^\infty(\mathbb{R})$, the question would be to characterise the torsion submodules of this $C^\infty$-module.
The only approach I've thought of so far, is maybe trying to understand what the "invariant factors" of this module are. By "invariant factors", I mean something like those that appear in the structure theorem for finitely-generated modules over a PID. The raging elephant in the room here of course is that $C^\infty (\mathbb{R})$ is not a PID, nor is $\mathcal{D} \, '$ finitely generated as a module over $C^\infty(\mathbb{R})$, so probably I'm asking for too much here.
Though, apart from this, I don't really see way forward. Any help would be much appreciated :)
