Algebra extension

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, [math]\displaystyle{ \phi }[/math]) consisting of a ring E and a ring homomorphism [math]\displaystyle{ \phi }[/math] that fits into the short exact sequence of abelian groups:

[math]\displaystyle{ 0 \to I \to E \overset{\phi}{{}\to{}} R \to 0. }[/math]^[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 [math]\displaystyle{ \phi }[/math] splits; i.e., [math]\displaystyle{ \phi }[/math] 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.

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

[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. Alternatively, E can be defined as [math]\displaystyle{ \operatorname{Sym}(M)/\bigoplus_{n \ge 2} \operatorname{Sym}^n(M) }[/math] where [math]\displaystyle{ \operatorname{Sym}(M) }[/math] is the symmetric algebra of M.^[3] 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. It is trivial since [math]\displaystyle{ r \mapsto (r, 0) }[/math] is a section (note this section is a ring homomorphism since [math]\displaystyle{ (1, 0) }[/math] is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to [math]\displaystyle{ R \oplus I }[/math] if [math]\displaystyle{ I^2 = 0 }[/math]. Indeed, identifying [math]\displaystyle{ R }[/math] as a subring of E using a section, we have [math]\displaystyle{ (E, \phi) \simeq (R \oplus I, p) }[/math] via [math]\displaystyle{ e \mapsto (\phi(e), e - \phi(e)) }[/math].^[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 [math]\displaystyle{ R/I }[/math]-bimodule.

More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient [math]\displaystyle{ R \to R_{\mathrm{red}} }[/math] of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

[math]\displaystyle{ 0 \to I^n/I^{n-1} \to R/I^{n-1} \to R/I^n \to 0 }[/math]

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

• 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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