Matsumura : Commutative Algebra Part 2 Daniel

1 Extension of a Ring by a Module Let C be a ring and N an ideal of C with N = 0. If C ′ = C/N then the C-module N has a canonical C ′-module structure. In a sense analogous with the notion of extension for modules, the data C,N is an “extension” of the ring C ′. Definition 1. Let C ′ be a ring and N a C ′-module. An extension of C ′ by N is a triple (C, ε, i) consisting of a ring C, a surjective ring morphism ε : C −→ C ′ and a morphism of C-modules i : N −→ C such that Ker(ε) is an ideal whose square is zero and the following sequence of C-modules is exact 0 // N i // C ε // C ′ // 0 Note that Ker(ε) has a canonical C ′-module structure, and i gives an isomorphism of C ′-modules N ∼= Ker(ε). Two extensions (C, ε, i), (C1, ε1, i1) are said to be isomorphic if there exists a ring morphism f : C −→ C1 such that ε1f = ε and fi = i1. That is, the following diagram of abelian groups commutes 0 // N i // C