Ring extension

In algebra, a ring extension of a ring R by an abelian group I is a pair (E, ϕ ) consisting of a ring E and a ring homomorphism ϕ that fits into the exact sequence of abelian groups:

Note I is then an ideal of E. Given a commutative ring A, an A-extension is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules".

An extension is said to be trivial if ϕ splits; i.e., ϕ admits a section that is an algebra homomorphism.

A morphism between extensions of R by I, over say A, is an algebra homorphism 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.