short exact sequence
A short exact sequence is the cleanest way to package the relationship between a submodule, a module, and the corresponding quotient into a single diagram. It says: here is a sub-object A sitting inside B, and C is exactly what is left when you collapse A. The whole of module theory is, in a sense, the study of how B is assembled out of the pieces A and C.
Formally it is a diagram of R-modules and homomorphisms 0 -> A -> B -> C -> 0 that is exact at every spot: the map A -> B is injective, the map B -> C is surjective, and the image of A -> B equals the kernel of B -> C. Thus A is (isomorphic to) a submodule of B, and C is isomorphic to the quotient B/A. Conversely every submodule A of B yields such a sequence with C = B/A.
The point of the formalism is that B is an 'extension' of C by A, and knowing A and C does not determine B: the same ends can be assembled in genuinely different ways. How many ways, and how they are classified, is measured by the group Ext(C, A). When the only extension is the trivial one B = A ⊕ C, the sequence splits; in general it need not, and the gap between general extensions and split ones is where homological algebra begins.
0 -> Z -> Z -> Z/2Z -> 0, where the first map is multiplication by 2 and the second is reduction mod 2. The middle Z is built from the sub-copy 2Z (≅ Z) and the quotient Z/2Z, but it is not their direct sum — this extension is non-split.
A non-split extension: Z is not Z ⊕ Z/2Z, so the ends do not determine the middle.