Module Theory From now on, R will denote an arbitrary ring with 1 ...

4
p1,...pk and positive integers e1,...,ek and an integer n ≥ 0 such that G is isomorphic to
⊕Zn .
Z p e 1
1 ⊕···⊕Z p e k
k
If n =0,thenGis finite. If k =0,thenGis free. In particular, every finitely generated
abelian group is a direct sum of cyclic groups.
Noetherian, Artinian modules and the Jordan-Hölder Theorem.
For this section, R is any ring with 1.
Theorem. Let M be a left R-module. The following are equivalent:
(1) All submodules of M are finitely generated.
(2) (ACC) Given any sequence of submodules N1 ⊆ N2 ⊆···, it must eventually become
constant.
(3) Any nonempty collection of submodules has a maximal element.
<strong>Module</strong>s satisfying the conditions of the previous theorem are called Noetherian. If all
left ideals of R are finitely generated, then R is a left Noetherian ring.
A module is called Artinian if the following hold
(1) (DCC) Every descending chain of submodules eventually becomes constant.
(2) Every collection of submodules has a minimal element.
These two conditions are equivalent by an analogous proof to that in the Noetherian case.
Theorem. Let N