This question has been completely reformulated following the guidelines in the comments below, to make it clearer where I was able to get and where I can't get out of. Such comments helped me a lot.
Let $(X,\mathscr{O}_{X})$ be a ringed space. Let $\mathscr{F}$ be a sheaf of $\mathscr{O}_{X}$-modules locally free.
Show that $\mathscr{F}$ is a quasi-coherent sheaf of $\mathscr{O}_{X}$-modules.
I've been thinking about the following: $\mathscr{F}$ locally free $\Rightarrow$ $X$ can be covered by open sets $U$ for which $\mathscr{F}|_{U}\cong \bigoplus_{i \in I}\mathscr{O}_{X}|_{U}$, for some set $I$.
As with any $ X $ scheme, the structure sheaf $\mathscr{O}_{X}$ is quasi-coherent, then $X$ can be covered by open affine subsets $U_j=$Spec $A_j$, such that for each $j$ there is an $A_j$-module $M_j$ with $\mathscr{O}_{X}|_{U_j} \cong \widetilde{M_j}$.
So, we have to: $\mathscr{F}|_{U \cap U_j}\cong \bigoplus_{i \in I}\mathscr{O}_{X}|_{U\cap U_j}\cong \bigoplus_{j \in I} \widetilde{M_j}\cong \widetilde{(\bigoplus_{j \in I} M_j)}$. Thus, $\mathscr{F}$is quasi-coherent.
But now I have a question. A quasi-coherent sheaf $\mathscr{F}$ is primarily a sheaf of $\mathscr{O}_X$-modules. So so that I can see that $\mathscr{O}_X$ is quasi-coherent, I have to look at $ \mathscr{O}_X(U)$ as $ (\mathscr{O}_X(U), +)$ a group and therefore a $\mathscr{O}_X(U)$ - module, for each open $U$ in $X$. To make sense of the definition of quasi-coherent in sheaf of rings $\mathscr{O}_{X} $.
Is that correct?