Inabstract algebra, analgebra extension is the ring-theoretic equivalent of agroup extension.

Precisely, aring extension of aringR by anabelian groupI is a pair (E,${\displaystyle \varphi }$) consisting of a ringE and aring homomorphism${\displaystyle \varphi }$ that fits into theshort exact sequence of abelian groups:

${\displaystyle 0\to I\to E\stackrel{\varphi }{\to }R\to 0.}$^[1]

This makesI isomorphic to atwo-sided ideal ofE.

Given acommutative ringA, anA-extension or anextension of anA-algebra is defined in the same way by replacing "ring" with "algebra overA" and "abelian groups" with "A-modules".

An extension is said to betrivial or tosplit if${\displaystyle \varphi }$ splits; i.e.,${\displaystyle \varphi }$ admits asection that is aring homomorphism^[2] (see§ Example: trivial extension).

Amorphism between extensions ofR byI, over sayA, is an algebra homomorphismE →E' that induces the identities onI andR. By thefive lemma, such a morphism is necessarily anisomorphism, and so two extensions are equivalent if there is a morphism between them.

LetR be a commutative ring andM anR-module. LetE =R ⊕M be thedirect sum of abelian groups. Define the multiplication onE by

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

Note that identifying (a,x) witha +εx where ε squares to zero and expanding out (a +εx)(b +εy) yields the above formula; in particular we see thatE is a ring. It is sometimes called thealgebra of dual numbers. Alternatively,E can be defined as${\displaystyle \mathrm{Sym}(M)/\underset{n\ge 2}{⨁}{\mathrm{Sym}}^n(M)}$ where${\displaystyle \mathrm{Sym}(M)}$ is thesymmetric algebra ofM.^[3] We then have the short exact sequence

${\displaystyle 0\to M\to E\stackrel{p}{\to }R\to 0}$

wherep is the projection. Hence,E is an extension ofR byM. It is trivial since${\displaystyle r\mapsto (r,0)}$ is a section (note this section is a ring homomorphism since${\displaystyle (1,0)}$ is the multiplicative identity ofE). Conversely, every trivial extensionE ofR byI is isomorphic to${\displaystyle R\oplus I}$ if${\displaystyle I^2=0}$. Indeed, identifying${\displaystyle R}$ as a subring ofE using a section, we have${\displaystyle (E,\varphi )\simeq (R\oplus I,p)}$ via${\displaystyle e\mapsto (\varphi (e),e-\varphi (e))}$.^[1]

One interesting feature of this construction is that the moduleM becomes an ideal of some new ring. In his bookLocal Rings,Nagata calls this process theprinciple of idealization.^[4]

This sectionneeds expansion. You can help byadding to it.(March 2023)

Especially indeformation theory, it is common to consider an extensionR of a ring (commutative or not) by an ideal whose square is zero. Such an extension is called asquare-zero extension, a square extension or just an extension. For a square-zero idealI, sinceI is contained in the left and right annihilators of itself,I is a${\displaystyle R/I}$-bimodule.

More generally, an extension by a nilpotent ideal is called anilpotent extension. For example, the quotient${\displaystyle R\to R_{\mathrm{r}\mathrm{e}\mathrm{d}}}$ of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

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

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.

• Formally smooth map
• TheWedderburn principal theorem, a statement about an extension by the Jacobson radical.

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

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

• Ring theory

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