A manifold is, essentially, something that "looks like" $\mathbb R^n$ locally. That is, a topological space $X$ such that for every $x\in X$ there is a continuous map $\varphi$ from an open set $U_x$ containing $x$ to $\mathbb R^n$.
Suppose I want to consider a possible generalization of this concept, in particular by extending which coordinate spaces I can use. A first naive generalization would be to admit any vector space $V$ instead of $\mathbb R^n$, however, any finite dimensional $V$ is isomorphic to some $\mathbb R^n$, so this is clearly not adding anything new. A possible next attempt would be to admit an arbitrary module $M$ (perhaps over a commutative ring) as the coordinate space.
My question would be: Is this concept one that's been explored? Could it produce an interesting object of study?