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 ϕ:E->R such that I is isomorphic (as an abelian group) to the kernel of ϕ. In other words,

0->I->E->ϕR->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 ϕ splits; i.e., ϕ 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.

## Contents

* 1 Examples
* 1.1 Example 1
* 1.2 Example 2
* 2 References

## Examples

### Example 1

Let's take the ring ℤ of whole numbers and let's take the abelian group ℤ2(under addition) of binary numbers. Let E = ℤ⊕ℤ2 we can identify multiplication on E by (x,a)⋅(y,b)=(xy,ϕ(x)a+ϕ(y)b)(where ϕ:ℤ->ℤ2 is the homomorphism mapping even numbers to 0 and odd numbers to 1). This gives the short exact sequence

0->ℤ->E->pℤ->0

Where p is the homomorphism mapping (x,a)↦aϕ(x).[disputed - discuss]

### 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)⋅(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->M->E->pR->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.[1]

## References

1. ↑ Nagata, Masayoshi (1962), Local Rings, Interscience Tracts in Pure and Applied Mathematics, 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN 0-88275-228-6

* E. Sernesi: Deformations of algebraic schemes

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