I am not super experienced in (commutative) algebra and in the course of some of my work I noticed that I had a need for morphisms of rings that are compatible with the asssociated module of derivations. Let me set some notation.
$ R $ and $ S $ are rings and and $ \phi \colon R \to S $ is a ring homomorphism. Let $ Der(R) $ and $Der(S)$ denote the respective modules of derivations. Since $ S $ is an $ R $-module, we can also define $ Der(R, S ) $ to be the $R$-module of derivations on $R$ with values in $S$.
Notice that there are natural $R$-module homomorphisms $ Der(R) \to Der(R,S) $ and $ Der(S) \to Der(R,S) $ by post- and pre-composition by $\phi$, respectively. There is also a natural $S$-module homomorphism $S \otimes_R Der(R) \to Der(R,S)$.
Now let me propose two definitions. My hope is that these definitions exist in the literature and I am just too ignorant to know the actual terminology.
Definition 1: We say that $\phi$ is of the first kind if there exists a unique morphism of $R$-modules $\alpha \colon Der(R) \to Der(S) $ which makes the following diagram commute:
$$ \begin{array}{ccc} Der(R) & \rightarrow & Der(S) \\ & \searrow & \downarrow \\ & & Der(R,S) \end{array} $$ Motivating Example: Suppose $M$ is a smooth manifold and $U$ is an open subset. The restriction map $C^\infty(M) \to C^\infty(U)$ is a ring homomorphism of the first kind. I suspect this is actually true for any locally ringed space (?).
Definition 2: We say that $\phi$ is of the second kind if there exists a unique morphism of $S$-modules $\beta \colon Der(S) \to S \otimes_R Der(R) $ which makes the following diagram commute: $$ \begin{array}{ccc} Der(S) & \rightarrow & S \otimes_R Der(R) \\ & \searrow & \downarrow \\ & & Der(R,S) \end{array} $$ Motivating Example: Suppose $M$ and $N$ are smooth compact manifolds and $f \colon M \to N $ is a smooth function. The pull-back map $f^* \colon C^\infty(N) \to C^\infty(M) $ is of the second kind.
With the definitions out of the way, let me ask my questions.
Question 1: Is there a nice category of rings where every homomorphism is of the first and/or second kind?
Question 2: Are the uniqueness conditions in the definitions necessary? In my fumbling efforts I have not been able to find counter-examples. (I have since partially answered this, see edit)
Question 3: Are morphisms of local rings always of the second kind?
Question 4: If I assume that $Der(R)$ and $Der(S)$ are projective and finitely generated modules does that imply that $\phi$ is of the second kind?
Question 5: When is a morphism of local rings of the first/second kind? In the algebraic geometry universe, which morphisms of schemes induce homomorphisms of the first/second kind?
EDIT 1: Regarding uniqueness I see that there are plenty of examples that fail uniqueness. For instance, if one takes $M$ to be a manifold and $\pi \colon M \times M \to M $ to be projection to the first factor, one can show that there are actually many module homomorphisms $ \mathcal{X}(M \times M) \to \mathcal{X} (M)$ which make the 'first kind' diagram commute. However, none of them are natural/canonical.
EDIT 2: I figured I should include the version in terms of Kahler forms. Let's assume that $R$ and $S$ are $k$-algebras for some fixed field $k$. We denote by $\Omega_R$ and $\Omega_S$ the usual modules of Kahler forms.
The universal property of the Kahler differential tells us that $$ Der(R) \cong Hom_R(\Omega_R,R) \quad Der(S) \cong Hom(\Omega_S,S) \quad Der(R,S) \cong Hom_R(\Omega_R, S) $$
One can use these isomorphisms to substitute these objects into the above diagrams.
The one glimmer of hope is that there is actually a natural homomorphism $\phi_* \colon \Omega_R \to \Omega_S$ obtained by the rule $ \phi_*(dr) = d(\phi(r))$. Still, this is not enough to get the maps. From this point of view, it seems likely that if one assumes that $\Omega_R$ and $\Omega_S$ are sufficiently nice modules, then $\phi$ will be of the first/second kind. It seems like it could also be related to the structure of $S$ as an $R$-module. Unfortunately, I haven't been able to show that these conditions suffice. I also have no intuition as to which sorts of rings will have projective/finitely generated/etc modules of Kahler forms.