$\mathbf{Field}$ subcategory of $\mathbf{Vec}$, $\mathbf{Ring}$ subcategory of $\mathbf{Mod}$? Since every field $\Bbbk$ is a vector space over itself, and every commutative ring $R$ with identity is a module over itself, is it true that one can embed $\mathbf{Field}$ into $\mathbf{Vec}$?
Consider the functor $F: \mathbf{Field} \longrightarrow \mathbf{Vec}$ defined by the mapping $R \mapsto M_R$, where $M_R$ is the (bi) module of $R$ over itself.
The question is fairly elementary, and my concern to why it would not be true is because this seems like an immensely useful result, yet I haven't seen anybody state it explicitly. 
To give a bit of (maybe unnessecary) context: The question comes from Noetherian rings, with their usual three equivalent definitions: satisfying the ascending chain condition, every non-empty partially ordered set of ideals has a maximum, and every ideal is finitely generated. Noetherian modules are defined similarly. My problem is that finitely generated is a property of modules, so one would have to "get out of"/"jump between" categories to make the last equivalence for rings. So my idea was to prove the equivalence for Noetherian modules and regard the equivalence for Noetherian rings as a special case.
Thank your very much in advance!
 A: The category of vector spaces over arbitrary fields mixes both morphisms of vector spaces and of the fields they’re over, and isn’t much used in elementary contexts, although it and its relatives have great applications later on. Anyway, it doesn’t work for this purpose, as described in the comments. Normally one uses the category of vector spaces over a fixed field. 
You could map the category of characteristic $p$ fields, where $p$ is prime or $0$, into the category of vector spaces over the corresponding prime field, $\mathbb{Z}/p$ or $\mathbb{Q}$ as the case may be, since every field morphism is linear over the prime field, and do something similar with rings. But this isn’t particularly interesting-it’s just a slight refinement of the embedding of these categories into abelian groups, which is certainly a bit of a yawn. More common, as was mentioned in the comments, is to study properties of a ring that arise from its canonical module structure over itself, which doesn’t involve embedding any categories.
