Algebra extension

Short description: Surjective ring homomorphism with a given codomain

In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension.

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

0->I->E->ϕR->0.[1]

This makes I isomorphic to a two-sided ideal of E.

Given a commutative ring A, an A-extension or an extension of an A-algebra 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 or to split if ϕ splits; i.e., ϕ admits a section that is a ring homomorphism[2] (see § Example: trivial extension).

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 Trivial extension example
* 2 Square-zero extension
* 3 See also
* 4 References
* 5 Further reading

## Trivial extension example

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. Alternatively, E can be defined as Sym⁡(M)/⨁n≥2Symn(M) where Sym⁡(M) is the symmetric algebra of M.[3] 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. It is trivial since r↦(r,0) is a section (note this section is a ring homomorphism since (1,0) is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to R⊕I if I2=0. Indeed, identifying R as a subring of E using a section, we have (E,ϕ)≃(R⊕I,p) via e↦(ϕ(e),e−ϕ(e)).[1]

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.[4]

## Square-zero extension

Especially in deformation theory, it is common to consider an extension R of a ring (commutative or not) by an ideal whose square is zero. Such an extension is called a square-zero extension, a square extension or just an extension. For a square-zero ideal I, since I is contained in the left and right annihilators of itself, I is a R/I-bimodule.

More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient R->Rred of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

0->In/In−1->R/In−1->R/In->0

is a square-zero extension. Thus, a nilpotent extension breaks up into successive square-zero extensions. Because of this, it is usually enough to study square-zero extensions in order to understand nilpotent extensions.

## See also

* Formally smooth map
* The Wedderburn principal theorem, a statement about an extension by the Jacobson radical.

## References

1. ↑ 1.0 1.1 Sernesi 2007, 1.1.1.
2. ↑ Typical references require sections be homomorphisms without elaborating whether 1 is preserved. But since we need to be able to identify R as a subring of E (see the trivial extension example), it seems 1 needs to be preserved.
3. ↑ Anderson, D. D.; Winders, M. (March 2009). "Idealization of a Module". Journal of Commutative Algebra 1 (1): 3–56. doi:10.1216/JCA-2009-1-1-3. ISSN 1939-2346. https://projecteuclid.org/journals/journal-of-commutative-algebra/volume-1/issue-1/Idealization-of-a-Module/10.1216/JCA-2009-1-1-3.full.
4. ↑ 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

* Sernesi, Edoardo (20 April 2007) (in en). Deformations of Algebraic Schemes. Springer Science & Business Media. ISBN 978-3-540-30615-3. https://books.google.com/books?id=xkcpQo9tBN8C.

## Further reading

* algebra extension at nLab
* infinitesimal extension at nLab
* Extension of an associative algebra at Encyclopedia of Mathematics

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