Noetherian module
A Noetherian module is a module in which you cannot keep enlarging submodules forever — every strictly increasing chain of submodules eventually stops. This is a finiteness condition that is much weaker than being finite or finitely generated as a set, yet strong enough to make many constructions terminate and many theorems work. It is the module-side analogue of a Noetherian ring.
An R-module M is Noetherian if it satisfies the ascending chain condition (ACC): every chain of submodules N_1 ⊆ N_2 ⊆ N_3 ⊆ ... is eventually constant. Three conditions are equivalent: (i) ACC on submodules; (ii) every nonempty collection of submodules has a maximal element; (iii) every submodule of M is finitely generated. The last form is usually the most useful in practice.
Noetherianness passes to submodules and quotients, and is closed under extensions: in a short exact sequence 0 -> A -> B -> C -> 0, B is Noetherian iff both A and C are. Consequently a ring R is Noetherian exactly when R is Noetherian as a module over itself, and then every finitely generated R-module is Noetherian. This is why finitely generated modules over Z, over a field, or over any polynomial ring k[x_1, ..., x_n] are all Noetherian (via the Hilbert basis theorem).
Z is a Noetherian Z-module: its submodules are 0 ⊂ ... ⊂ 4Z ⊂ 2Z ⊂ Z, and any ascending chain nZ ⊆ mZ requires m | n, so divisibility forces the chain to stabilize. By contrast the polynomial ring k[x_1, x_2, ...] in infinitely many variables, as a module over itself, is not Noetherian.
ACC holds in Z but fails for infinitely many variables, the standard non-Noetherian example.