Reading a short exact sequence
A sequence of modules and maps is exact at a spot when image = kernel there. A short exact sequence 0 → A → B → C → 0 packs three statements at once: A → B is injective (so A is a submodule of B), B → C is surjective, and C ≅ B/A. Read it as “B is built from a sub-module A and a quotient C.” Almost every structural question about B becomes: how is B assembled from its piece A and its piece C?
Two short exact sequences with the SAME ends but different middles.
(1) 0 -> Z/2Z -> Z/4Z -> Z/2Z -> 0
inclusion {0,2} ↪ Z/4Z, then Z/4Z ↠ Z/4Z over {0,2} = Z/2Z.
Middle is Z/4Z: a single cyclic group, NOT a direct sum.
(2) 0 -> Z/2Z -> Z/2Z (+) Z/2Z -> Z/2Z -> 0
include into first factor, project to second.
Middle is the Klein four-group (Z/2Z)^2.
Same A = C = Z/2Z on the ends, but Z/4Z is NOT isomorphic to (Z/2Z)^2
(one has an element of order 4, the other does not). So knowing A and C
does NOT determine B. The difference is whether the sequence SPLITS:
(2) splits, (1) does not. That single bit is the extension problem.When does it split?
A short exact sequence is split when B ≅ A ⊕ C in the compatible way — equivalently, when the surjection B → C has a section (a right inverse), or the injection A → B has a retraction (a left inverse). The splitting lemma says these conditions are equivalent. Splitting is exactly the difference between examples (1) and (2) above. A clean sufficient condition: if C is projective, every short exact sequence ending in C splits — indeed this is one definition of projective. Dually, if A is injective, every sequence starting at A splits.
Noetherian and Artinian: taming the infinite
A module is Noetherian if it satisfies the ascending chain condition (every increasing chain of submodules stabilizes), equivalently every submodule is finitely generated. It is Artinian if it satisfies the descending chain condition. The key lemma: in a short exact sequence 0 → A → B → C → 0, B is Noetherian (resp. Artinian) iff both A and C are. So these properties pass to submodules, quotients, and extensions — exactly the closure you want.
A module is both Noetherian and Artinian iff it has a composition series: a chain 0 = M₀ ⊂ M₁ ⊂ ⋯ ⊂ Mₙ = M with each quotient Mᵢ/Mᵢ₋₁ simple. The Jordan-Hölder theorem then guarantees the length n and the multiset of simple quotients are invariants of M, independent of the chosen series — the module analogue of factoring an integer into primes. Note Z is Noetherian but not Artinian (the chain Z ⊋ 2Z ⊋ 4Z ⊋ ⋯ never stops), so it has no composition series, fitting its infinite size.