First at all i have to say I am very new to categories (just basic definition).
My Question:- We have two categories ( $\mathcal{C},\mathcal{R}$) object in both of these two categories are module ( defined over infinite-dimensional algebra), we can embed the first one into the second, so we have a functor, $ F:\mathcal{C} \rightarrow \mathcal{R}$. my question is there is a natural way to induce a adjoint functor ? ( We can look at the first categories to be subcategories of the second). again, I am very new and any help appreciated.
The answer to your particular question is that there's no such adjunction in general. Subcategories for which the inclusion has a left (right) adjoint are called (co)reflective, and are very common but far from universal. Vaguely, you need an optimal way to round objects of the bigger category down to objects of the smaller category, and there's no general way to do this. The SAFT and GAFT I mention in my comment are the best known general ways of checking when you can. For categories of modules (if you're taking all modules over some algebra at once) their technical assumptions are automatically satisfied, so we're left with the claim that the inclusion is a left adjoint if and only if it preserves all colimits, and similarly for a right adjoint.