Splitting of Short Exact Sequences for Modules

(1.1) 0 −→ N f −−→M g −−→ P −→ 0 which is exact at N , M , and P . That means f is injective, g is surjective, and im f = ker g. Example 1.1. For an R-module M and submodule N , there is a short exact sequence 0 // N // M // M/N // 0, where the map N →M is the inclusion and the map M →M/N is reduction modulo N . Example 1.2. For R-modules N and P , the direct sum N ⊕ P fits into the short exact sequence 0 // N // N ⊕ P // P // 0, where the map N → N ⊕ P is the embedding n 7→ (n, 0) and the map N ⊕ P → P is the projection (n, p) 7→ p. Example 1.3. Let I and J be ideals in R such that I+J = R. Then there is a short exact sequence 0 // I ∩ J // I ⊕ J + // R // 0, where the map I ⊕ J → R is addition, whose kernel is {(x,−x) : x ∈ I ∩ J}, and the map I∩J → I⊕J is x 7→ (x,−x). This is not the short exact sequence 0 −→ I −→ I⊕J −→ J −→ 0 as in Example 1.2, even though the middle modules in both are I ⊕ J . Any short exact sequence that looks like the short exact sequence of a direct sum in Example 1.2 is called a split short exact sequence. More precisely, a short exact sequence 0 −→ N f −−→ M g −−→ P −→ 0 is called split when there is an R-module isomorphism θ : M → N ⊕ P such that the diagram