Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e. a ring), we can associate to it a category of modules $R-Mod$; namely, an object $(N, \alpha)$ in $R-Mod$ is the data of an object $N$ in $\mathcal{C}$ and a morphism $\alpha: R \otimes N \rightarrow N$ in $\mathcal{C}$ such that $$ \alpha \circ (\mu_R \otimes 1_N ) = \alpha \circ (1_R \otimes \alpha) : R \otimes R \otimes N \rightarrow N,$$ $$ \alpha \circ \{1_R\} \otimes 1 = 1 : N \rightarrow N.$$ There is a forgetful functor $$U : R-Mod \rightarrow \mathcal{C}$$ with its left adjoint $$F: \mathcal{C} \rightarrow R-Mod.$$ If $R$ is a field, then the objects in the image of $F$ are exactly all the $R$-modules and this can be proven using the axiom of choice since they are $R$-vector spaces. So I ask::
Is there a way to prove this fact using only properties of the fields as objects in the category $\mathcal{C}$?
Are there other properties of fields or rings in general that can be viewed in this context (for example the properties of being artinian of noetherian, or the characteristic of a field)?