Noetherian module

In algebra, a Noetherian module is a module with a condition on the lattice of submodules.

Definition
Fix a ring R and let M be a module. The following conditions are equivalent:


 * 1) The module M satisfies an ascending chain condition on the set of its submodules: that is, there is no infinite strictly ascending chain of submodules $$M_0\subsetneq M_1\subsetneq M_2\subsetneq\ldots$$.
 * 2) Every submodule of M is finitely generated.
 * 3) Every nonempty set of submodules of M has a maximal element when considered as a partially ordered set with respect to inclusion.

When the above conditions are satisfied, M is said to be Noetherian.

Examples

 * A zero module is Noetherian, since its only submodule is itself.
 * A Noetherian ring (satisfying ACC for ideals) is a Noetherian module over itself, since the submodules are precisely the ideals.
 * A free module of finite rank over a Noetherian ring is a Noetherian module.
 * A finite-dimensional vector space over a field is a Northerian module.
 * A free module of infinite rank over an infinite set is not Noetherian.