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 [math]\displaystyle{ \phi:E\to R }[/math] such that I is isomorphic (as an abelian group) to the kernel of [math]\displaystyle{ \phi. }[/math] In other words,

[math]\displaystyle{ 0 \to I \to E \overset{\phi}{{}\to{}} R \to 0 }[/math]

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 [math]\displaystyle{ \phi }[/math] splits; i.e., [math]\displaystyle{ \phi }[/math] 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 [math]\displaystyle{ \mathbb Z }[/math] of whole numbers and let's take the abelian group [math]\displaystyle{ \mathbb Z_2 }[/math](under addition) of binary numbers. Let E = [math]\displaystyle{ \mathbb Z \oplus \mathbb Z_2 }[/math] we can identify multiplication on E by [math]\displaystyle{ (x,a) \cdot (y,b) = (xy,\phi(x)a+\phi(y)b) }[/math](where [math]\displaystyle{ \phi:\mathbb Z \to \mathbb Z_2 }[/math] is the homomorphism mapping even numbers to 0 and odd numbers to 1). This gives the short exact sequence

[math]\displaystyle{ 0\to \mathbb Z \to E \overset{p}{{}\to{}} \mathbb Z \to 0 }[/math]

Where p is the homomorphism mapping [math]\displaystyle{ (x,a) \mapsto a\phi(x) }[/math].

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

[math]\displaystyle{ (a, x) \cdot (b, y) = (ab, ay + bx). }[/math]

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

[math]\displaystyle{ 0 \to M \to E \overset{p}{{}\to{}} R \to 0 }[/math]

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