TITLE: Idea behind defining a Projective System
QUESTION [6 upvotes]: What is the idea behind defined a Projective system of Groups/Rings. In our class an example for the Projective system was given by Taking the Ring $\mathbb{Z}/n\mathbb{Z}$ over $\mathbb{N}$. The order defined was: 

$ m \leq n$ if $m \mid n$.

Moreover then we calculated the Projective Limit of this system and found it to be $\hat{\mathbb{Z}}$. I have not at all understood this concept fully so even while posing this question there might be errors. So people who can understand what i am talking about are welcome to pose answers. If one still has problems with the question please let me know i shall try to improve it or delete it.

REPLY [7 votes]: Basically, a projective system formalizes the idea of "constructing elements by successive approximations" in an algebraic setting.
If $G$ is a group (abelian, to fix ideas) and $G_n$ a sequence of smaller and smaller subgroups, the projective limit $\Gamma=\lim(G/G_n)$ is the set of elements that are defined by giving them up to an ambiguity that gets smaller and smaller.
When $G$ is also a topological space and the $G_n$ are a local basis around the identity element, an element of $\Gamma$ is morally an element given by assigning a Cauchy sequence converging to it.
Incidentally, this is exactly the case for the $p$-adic integers ${\Bbb Z}_p$ because it is possible to define a metric in $\Bbb Z$ in such a way the ideal $p^n{\Bbb Z}$ are open spheres.