*Note: For the record, exterior algebras and derivations are irrelevant to my question. However, I have a hard time assessing what I want to ask and I find it is the easiest to do so using a direct example. Hence exterior algebras.*

When studying derivations of the exterior algebra over a smooth manifold $M$, $\Omega(M)$, one often encounters statements such as

A derivation is local, eg. for $\omega\in\Omega(M)$, $(D\omega)|_U=D(\omega|_U)$, where $D$ is a derivation of the exterior algebra, and $U$ is an open set of $M$.

A derivation is algebraic if it vanishes on the grade 0 subspace. An algebraic derivation is ultralocal, in the sense $D\omega|_p$ depends only on $\omega_p$ if $D$ is an algebraic derivation.

Many of these statements reference or use the fact that $\Omega(M)$ is an associative, anticommutative $\mathbb{R}$-, and $C^\infty(M)$-algebra, that is also $\mathbb{N}$-graded, however for example the two I singled out rely on the fact that an element of $\Omega(M)$ is a **map** $M\rightarrow\wedge T^*M$ (that is also a section).

If I wanted to abstract studying derivations of the exterior algebra into studying derivations of an $R$-algebra ($R$ is a commutative ring) that is associative, anticommutative and $\mathbb{N}$-graded, I could do it, but then statements such as "$D$ is local" or "$D$ is ultralocal" would make no sense, since if $\omega$ is an element of this algebra, things such as $\omega|_U$ or $\omega_p$ don't make sense.

I realize that the exterior algebra has two additional operations in this case, a "restriction", which maps a pair $(U,\omega)$ ($U$ is open and $\omega$ is a differential form on $M$, *but it can also be a differential form defined on any region that contains $U$*) to $\omega|_U\in\Omega(U)$ and an "evaluation", which maps the pair $(p,\omega)$ (where $p\in M$ and $\omega$ is once again either a form on $M$ or a form defined on a region that contains $U$) to $\omega_p\in\wedge T_p^*M$, but it is even hard for me to come up with a rigorous description of these maps. I'll attempt:

Evaluation is the following map: $$\text{Eval}:M\times\Omega(M)\rightarrow\wedge T^ *M$$ such that $\text{Pr}_1(M\times\Omega(M))=\pi_{\wedge T^*M}\circ\text{Eval}$, where $\text{Pr}_1(M\times\Omega(M))$ is the projection to the first factor.

Let the "space of local differential forms" be $\Omega_{\text{loc}}(M)=\bigsqcup_{U\in\tau_M}\Omega(U)$, so an element of $\Omega_{\text{loc}}(M)$ is a pair $(U,\omega)$ where $U\in\tau_M$ is open and $\omega$ is defined on $U$ only. The restriction is then a map $$ \text{Res}:\tau_M\times\Omega(M)\rightarrow\omega_{\text{loc}}(M) $$ such that $\text{Pr}_1(\tau_M\times\Omega(M))=\text{Dom}\circ\text{Res}$ and $\text{Eval}(p,\cdot)\circ\text{Res}=\text{Eval}(p,.)\circ\text{Pr}_2(\tau_M\times\Omega(M))$ where $\text{Eval}$ has been extended to a partial function on $\Omega_{\text{loc}}(M)$ and $p$ is an arbitrary point of $M$ for which "Eval" as a partial function is defined.

I have no idea whether my "Res" and "Eval" functions capture the "locality" of the exterior algebra well, and although no references are made to elements of the exterior algebra being functions, I have referenced $M$, even though for a general $R$-algebra, I'd have no access to $M$.

So basically, my question is, how can I tell if an $R$-algebra/ring I am studying in the abstract can be realized as some kind of algebra of maps? Is there a name for such algebras/rings?