There seems to be a strange phenomenon occuring in several areas of mathematics.
In group theory, one may quotient away the commutator subgroup to obtain an abelian group.
In the theory of orders, one may collapse equivalence classes (defined by $x \sim y :\Leftrightarrow x \le y \wedge y \le x$) to obtain a (total) order from a (total) preorder.
There are more examples like this: In ring theory for instance, one can divide the nilradical to obtain a ring without nilpotent elements (one can also divide by maximal/prime ideals, but here I don't see how the choice would be unique from the ring).
The phenomenon we observe is this: By collapsing away a canonically defined "unpleasant" subspace (that is one with elements that should not exist in the structure we want), we obtain a "nicer" space (which may contain less information, but may allow (a simplification of) calculations that only rely on the essential structure which is preserved.
My question is this: Are there other places in mathematics where this occurs? Where? How does the construction go?
A further question would be: Is there a category-theoretic interpretation of this phenomenon?