# Constructions of Rings and Modules
# Free Modules and Rings
Intuitively, "free" objects in a category are constructed in the most general way possible, without extra conditions, hence they "lie" above all over objects.
# Free Modules
To elucidate further on the concept of an object being "free", we first shall consider a module freely generated by a set, since it has no superfluous condition in its construction.
A base for an -module is a subset of such that every element of has the form
where almost all of the are zero and are uniquely determined. A module with a base is called free.
- is free in , with base (which has elements).
- The ring of triangular matrices over is free in , with base (which has elements).
- is a free -module, with base .