An extension is said to be trivial if splits; i.e., admits a section that is an algebra homomorphism.
A morphism between extensions of R by I, over say A, is an algebra homomorphism E → E' that induces the identities on I and R. By the five lemma, such a morphism needs to be an isomorphism and two extensions are equivalent if there is a morphism between them.
Example: Let R be a commutative ring and M an R-module. Let E = R ⊕ M be the direct sum of abelian groups. Define the multiplication on E by
Note identifying (a, x) with a + εx, where ε squares to zero, and expanding (a + εx)(b + εy) out yield the above formula; in particular, we see E is a ring. We then have the exact sequence
where p is the projection. Hence, E is an extension of R by M. One interesting feature of this construction is that the module M becomes an ideal of some new ring. In his "local rings", Nagata calls this process the principle of idealization.
- E. Sernesi: Deformations of algebraic schemes
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