Modules
The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to for some .
If M is a free module over a principal ideal domain R, then every submodule of M is again free. This does not hold for modules over arbitrary rings, as the example of modules over shows.
Read more about this topic: Principal Ideal Domain