Ring extension

Short description: Surjective ring homomorphism with a given codomain

In commutative algebra, a ring extension of a ring R by an abelian group I is a pair of a ring E and a surjective ring homomorphism $\varphi :E\to R$ such that I is isomorphic (as an abelian group) to the kernel of $\varphi .$ In other words,

$0\to I\to E\stackrel{\varphi }{\to }R\to 0$

is a short exact sequence of abelian groups. (This makes I 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 $\varphi$ splits; i.e., $\varphi$ admits a section that is a rng homomorphism. This implies that E is isomorphic to the direct product of R and I.

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 $\mathbb{\mathbb{Z}}$ of whole numbers and let's take the abelian group ${\mathbb{\mathbb{Z}}}_2$(under addition) of binary numbers. Let E = $\mathbb{\mathbb{Z}}\oplus {\mathbb{\mathbb{Z}}}_2$ we can identify multiplication on E by $(x,a)\cdot (y,b)=(xy,\varphi (x)a+\varphi (y)b)$(where $\varphi :\mathbb{\mathbb{Z}}\to {\mathbb{\mathbb{Z}}}_2$ is the homomorphism mapping even numbers to 0 and odd numbers to 1). This gives the short exact sequence

$0\to \mathbb{\mathbb{Z}}\to E\stackrel{p}{\to }\mathbb{\mathbb{Z}}\to 0$

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

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. It is sometimes called the algebra of dual numbers. We then have the short exact sequence

$0\to M\to E\stackrel{p}{\to }R\to 0$

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 book Local Rings, Nagata calls this process the principle of idealization.

References

• E. Sernesi: Deformations of algebraic schemes