Ring extensions (Algebra)

In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension. Precisely, a ring extension of a ring R by an abelian group I is a pair (E, ϕ {\displaystyle \phi } ) consisting of a ring E and a ring homomorphism ϕ {\displaystyle \phi } that fits into the short exact sequence of abelian groups: 0 → I → E → ϕ R → 0. {\displaystyle 0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.} This makes I isomorphic to a two-sided ideal of E. Given a commutative ring A, an A-extension or an extension of an A-algebra 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 or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism (see § Example: trivial extension). 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 is necessarily an isomorphism, and so two extensions are equivalent if there is a morphism between them.