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 consisting of a ring E and a ring homomorphism that fits into the short exact sequence of abelian groups:
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 splits; i.e., admits a section that is a ring homomorphism.
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
Note that identifying with a + εx where ε squares to zero and expanding out 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 where is the symmetric algebra of M. We then have the short exact sequence
where p is the projection. Hence, E is an extension of R by M. It is trivial since is a section. Conversely, every trivial extension E of R by I is isomorphic to if. Indeed, identifying as a subring of E using a section, we have via.
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 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 -bimodule.
More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient of a Noetherian commutative ring by the nilradical is a nilpotent extension.
In general,
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.