# Ring extension

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

Note *I* is then a two-sided 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 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 is necessarily an isomorphism, and so 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 that identifying (*a*, *x*) with *a* + *εx* where ε squares to zero and expanding out (*a* + *εx*)(*b* + *εy*) yields the above formula; in particular we see that *E* is a ring. We then have the short 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*.