Ring extension

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

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

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 $\phi$ splits; i.e., $\phi$ 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.

Examples

Example 1

Let's take the ring $\mathbb {Z}$ of whole numbers and let's take the abelian group $\mathbb {Z} _{2}$ (under addition) of binary numbers. Let E = $\mathbb {Z} \oplus \mathbb {Z} _{2}$ we can identify multiplication on E by $(x,a)\cdot (y,b)=(xy,\phi (x)a+\phi (y)b)$ (where $\phi :\mathbb {Z} \to \mathbb {Z} _{2}$ is the homomorphism mapping even numbers to 0 and odd numbers to 1). This gives the short exact sequence

0\to \mathbb {Z} \to E{\overset {p}{{}\to {}}}\mathbb {Z} \to 0

Where p is the homomorphism mapping $(x,a)\mapsto a\phi (x)$ .

Example 2

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

(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

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