Limits, Colimits and How to Calculate Them in the Category of Modules over a Pid

The goal of this paper is to introduce methods that allow us to calculate certain limits and colimits in the category of modules over a principal ideal domain. We start with a quick review of basic categorical language and duality. Then we develop the concept of universal morphisms and derive limits and colimits as special cases. The completeness and cocompleteness theorems give us methods to calculate the morphisms associated with limits and colimits in general. To use these methods, we then specialize to the case of finitely generated modules over a PID. We first develop the Smith normal form as a computational tool and prove the structure theorem for finitely generated modules over a PID. Lastly, we discuss how to use the abstract methods suggested by the completeness and cocompleteness theorems in the context of RMod.