Algebra extension

For the ring-theoretic equivalent of a field extension, see Subring#Ring extensions.

Not to be confused with Algebraic 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, $\phi$) consisting of a ring E and a ring homomorphism $\phi$ that fits into the short exact sequence of abelian groups:

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

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 $\phi$ splits; i.e., $\phi$ admits a section that is a ring homomorphism (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

$(a,x)\cdot (b,y)=(ab,ay+bx).$

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. It is sometimes called the algebra of dual numbers. Alternatively, E can be defined as $\operatorname {Sym} (M)/\bigoplus _{n\geq 2}\operatorname {Sym} ^{n}(M)$ where $\operatorname {Sym} (M)$ is the symmetric algebra of M. 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. It is trivial since $r\mapsto (r,0)$ 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 $R\oplus I$ if $I^{2}=0$. Indeed, identifying $R$ as a subring of E using a section, we have $(E,\phi )\simeq (R\oplus I,p)$ via $e\mapsto (\phi (e),e-\phi (e))$.

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.

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 $R\to R_{\mathrm {red} }$ of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

$0\to I^{n}/I^{n-1}\to R/I^{n-1}\to R/I^{n}\to 0$

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

• Sernesi, Edoardo (20 April 2007). Deformations of Algebraic Schemes. Springer Science & Business Media. ISBN 978-3-540-30615-3.

Further reading

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