In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension.
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
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.
Let E = R ⊕ M be the direct sum of abelian groups.
Define the multiplication on E by 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.
is the symmetric algebra of M.[3] We then have the short exact sequence where p is the projection.
Conversely, every trivial extension E of R by I is isomorphic to
[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] 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.
For a square-zero ideal I, since I is contained in the left and right annihilators of itself, I is a
More generally, an extension by a nilpotent ideal is called a nilpotent extension.
of a Noetherian commutative ring by the nilradical is a nilpotent extension.