Ring extension

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

${\displaystyle 0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.}$

This article is about the ring-theoretic equivalent of a group extension. For the ring-theoretic equivalent of a field extension, see Subring § Ring extensions.

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 ${\displaystyle \phi }$ splits; i.e., ${\displaystyle \phi }$ 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

${\displaystyle (a,x)\cdot (b,y)=(ab,ay+bx).}$

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

${\displaystyle 0\to M\to E{\overset {p}{{}\to {}}}R\to 0}$

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.

References

• E. Sernesi: Deformations of algebraic schemes