Let be a ring and a left -module. Then we say that is a Noetherian module if it satisfies the following property, known as the ascending chain condition (ACC):
- For any ascending chain
- of submodules of , there exists an integer so that (i.e. the chain eventually stabilizes, or terminates).
We say that a ring is left (right) Noetherian if it is Noetherian as a left (right) -module. If is both left and right Noetherian, we call it simply Noetherian.
Theorem. The following conditions are equivalent for a left -module:
- is Noetherian.
- Every submodule of is finitely generated (i.e. can be written as for some ).
- Every collection of submodules of has a maximal element.
The second condition is also frequently used as the definition for Noetherian.
We also have right Noetherian modules with the appropriate adjustments.
Proof. In general, condition 3 is equivalent to ACC. It thus suffices to prove that condition 2 is equivalent to ACC.
Suppose that condition 2 holds. Let be an ascending chain of submodules of . Then is a submodule of , so it must be finitely generated, say by elements . Each of the is contained in one of , say in . If we set , then for all , so Thus satisfies ACC.
On the other hand, suppose that condition 2 does not hold, that there exists some submodule of that is not finitely generated. Thus we can recursively define a sequence of elements such that is not in the submodule generated by . Then the sequence is an ascending chain that does not stabilize.
Note: The notation denotes the module generated by .
Hilbert's Basis Theorem guarantees that if is a Noetherian ring, then is also a Noetherian ring, for finite . It is not a Noetherian -module.