In the last post, we discussed the quasi-coherent sheaves for some ring $R$ and scheme $X$. On a scheme $X$, the quasi-coherent sheaves form an abelian category and in fact, this category is a sub-category of the category of $R$-modules. So, simply, as Vakil puts it in his notes, one should better look if the category of $R$-modules is an abelian category and prove that the category of quasi-coherent sheaves (call $Q_{coh}$) is indeed a subcategory of the category of $R-$modules ($Mod_R$), so
$$Q_{coh} \subset Mod_R$$
I will leave this to you to prove this.
Similarly, the coherent sheaves for some ring $R$ also form the abelian category (similar proof) and in fact, quasi-coherent sheaves will not always form an abelian category for any arbitrary ringed space while coherent sheaves will always form an abelian category. Coherent sheaves come with a bit more than quasi-coherent sheaves, both attached very strongly to the sheaf of modules of $R$. The extra condition is of finite presentation and finitely generated modules. A quasi-coherent sheaf is a coherent sheaf if the modules $M$ are finitely generated (hence $R^n \to M$ which is a surjection). For the Noetherian scheme, a finitely generated quasi-coherent sheaf will automatically be a coherent sheaf. But for non-Noetherian schemes, it is not guaranteed. That is why, one should be careful defining coherent sheaf as quasi-coherent sheaf which is finitely generated, which is not true always.
