I have the following exercise
Let $M,N$ be open subsets in $\mathbb{R}^{n}$ and let $\Psi:M\to N$ be a smooth map. Show that $\Psi$ defines a morphism of spaces ringed by smooth functions.
The definitions are the following:
A presheaf $\mathcal{F}$ of functions on a topological space $M$ is a collection of subrings $\mathcal{F}(U)\subset C(U)$ of the ring $C(U)$ of all functions on $U$, for each open subset $U\subset M$, such that the restriction of every $\gamma\in\mathcal{F}(U)$ to an open subset $U_{1}\subset U$ belongs to $\mathcal{F}(U_{1})$.
A sheaf $\mathcal{F}$ of functions on a topological space $M$ is a presheaf on $M$ such that the following condition is met. Let $\{U_{i}\}_{i\in I}$ be an open cover (possibly infinite) of an open subset $U\subset M$ and $f_{i}\in\mathcal{F}(U_{i})$ a family of functions defined on the open sets of the cover, compatible on the pairwise intersections: $$f_{i}\left.\right\vert_{U_{i}\cap U_{j}}=f_{j}\left.\right\vert_{U_{i}\cap U_{j}} $$ for every pair $(U_{i},U_{j})$ of members of the cover. Then, there exists $f\in\mathcal{F}(U)$ such that $f_{i}$ is the restriction of $f$ to $U_{i}$ for all $i\in I$.
A ringed space $(M,\mathcal{F})$ is a a topological space equipped with a sheaf of functions.
A morphism $(M,\mathcal{F}_{M})\stackrel{\Psi}{\longrightarrow}(N,\mathcal{F}_{N})$ of ringed spaces is a continuous map such that, for every open subset $U\subset N$ and every function $f\in\mathcal{F}_{N}$, the function $f\circ\Psi$ belongs to the ring $\mathcal{F}_{M}(\Psi^{-1}(U))$.
I do not understand how to proof that a particular function (here, $f\circ\Psi$) belongs to some ring. Here, it is clear that $f\circ\Psi$ is smooth, as the composite of smooth functions. But what other conditions do I have to show? What does that mean to "belong to the ring $\mathcal{F}_{M}(\Psi^{-1}(U))$"?
The ring $\mathcal{F}(U)$ is equipped with which operations? I suppose they are usual addition and multiplication of functions. But I guess this exercise is not only about showing that $f\circ\psi^{-1}+g$ and $(f\circ\psi^{-1})\cdot g$ are both smooth functions in $\mathcal{F}_{M}(\Psi^{-1}(U))$ for any $g\in\mathcal{F}_{M}(\Psi^{-1}(U))$ for all open $U\subset N$.
To summarize my question: what do I need to show?
