Disclaimer: This question is not transcendental at all, so go easy on me.
Starting with a (for simplicity) commutative unital ring $R$, we define a $R$-module. Obviously, since the name of the ring was italicized, the same must be done each time we write “$R$-module”, but if we are talking about the general theory of modules over a ring, we simply write “module”.
On the other hand, given a ring $R$ of differential operators, we call a module over $R$ a D-module (“D” stands for “differential”, I guess$\ldots$). Of course, to denote the ring by $R$ in this situation is strange, so common sense dictates that we must use a “mathematical” variant of the letter D to denote such ring ($D,\mathscr{D},\mathcal{D}$, whatever$\ldots$). This subclass of modules is of great importance, so the corresponding theory has a name of its own: D-module theory, right?
What is, then, the correct choice when we refer to the theory of modules over a ring of differential operators: “D-module theory” or “$D$-module theory”?
I am working with $D$-modules and I as you can see I am using the italic version. That is because it feels more natural to use the italic version as for other rings $R$ and $R$-modules. I am only working with one specific ring of differential operators though, which is the Weyl-Algebra $D = D_n$. So for me a $D$-module is really just a module over a specific ring and not a general module over some ring of differential operators, but I would also use the italic version in the latter case.