writing a field as an R module let $F$ be a field. for which ring $R$, $F$ is an $R$-module. i know already that as an abelian group $F$ is a $\mathbb Z$- module, what else can we say for a general field $F$. 
 A: I'm afraid there isn't an awful lot we can say in general. The way the question was posed, the field multiplication is not tied to the $R$-module action at all. Therefore any kind of a module action on the additive group will work. For example, if $F=\mathbf{Q}(i,\sqrt{5})$, then additively $F$ is isomorphic to the Cartesian power $\mathbf{Q}^4$, and therefore $F$ can be viewed as a module over any ring that has a 4-dimensional rational representation, e.g. the ring of rational Hamiltonian quaternions. Similarly $F$ can be turned into a $\mathbf{Q}[G]$-module for a group $G$ that is isomorphic to any $\mathbf{Q}[G]$-module
$V$ with $\dim_{\mathbf{Q}}V=4$ simply by defining the module action in such a way that a linear isomorphism between $V$ and $F$ becomes a module isomorphism. 
The characteristic of the field will place some restrictions. If $F$ has characteristic zero, then it has no additive torsion, and the ring $R$ cannot have any torsion either. If $F$ has characteristic $p$, then the additive order of $1_R$ must be either infinite or a multiple of $p$. In other words $F$ will always be either a $\mathbf{Q}$-module (when $char F=0$) or a $\mathbf{Z}/p\mathbf{Z}$-module (when $char F=p$). 
Similarly, as you have seen in a field theory course, $F$ is always a module over any of its subfields.
The question would become quite a bit more interesting, if you tie the $R$-action to the $F$-multiplication. A natural way of doing that would be to insist that for all $x\in F$ the mapping $\rho_x:F\rightarrow F, a\mapsto ax$ would also be a homomorphism of $R$-modules. IOW, $F$ would be what is known as a $(R,F)$-bimodule. This places more severe constraints on the ring $R$.
But before we attempt anything like that I ask you to comment. Is this anything like what you expect? I am uncertain as to exactly what you want to learn? I understood the question to mean that given a field $F$, what kind of rings $R$ act on it in a way that turn the additive group $(F,+)$ into an $R$-module. If you wanted to ask something else, please be more specific :-)
