Ring extension

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.

In commutative algebra, a ring extension is a ring homomorphism $R\to S$ of commutative rings, which makes S an R-algebra.

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

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

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 $\phi$ splits; i.e., $\phi$ admits a section that is an algebra 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 $\mathbb {Z}$ 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 book Local Rings, Nagata calls this process the principle of idealization.

References

• E. Sernesi: Deformations of algebraic schemes