Artinian module
An Artinian module is the descending mirror of a Noetherian one: you cannot keep shrinking submodules forever — every strictly decreasing chain of submodules eventually stops. Where Noetherian forbids growing without end, Artinian forbids shrinking without end, and the two conditions, though formally dual, behave very differently in practice.
An R-module M is Artinian if it satisfies the descending chain condition (DCC): every chain N_1 ⊇ N_2 ⊇ N_3 ⊇ ... is eventually constant; equivalently, every nonempty collection of submodules has a minimal element. As with the ascending version, Artinianness passes to submodules and quotients and is closed under extensions.
A module is both Noetherian and Artinian exactly when it has a composition series — a finite chain of submodules with simple successive quotients — and then the Jordan-Holder theorem makes the length and the multiset of composition factors invariants of the module. The asymmetry between the two chain conditions is striking: Z is Noetherian but not Artinian (the chain Z ⊇ 2Z ⊇ 4Z ⊇ ... never stops), and for rings the Hopkins-Levitzki theorem shows that a left Artinian ring is automatically left Noetherian, though no such implication holds for modules in general.
The Prüfer group Z(p^∞) (all p-power roots of unity) is an Artinian Z-module that is not Noetherian: its proper submodules form a single ascending chain 0 ⊂ Z/p ⊂ Z/p^2 ⊂ ..., so descending chains terminate but ascending ones do not — the exact opposite of Z.
The Prüfer group is Artinian but not Noetherian; Z is the reverse — the conditions are independent.