Algebra extension

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

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

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 ${\displaystyle \operatorname {Sym} (M)/\bigoplus _{n\geq 2}\operatorname {Sym} ^{n}(M)}$ where ${\displaystyle \operatorname {Sym} (M)}$ is the symmetric algebra of M.[3] We then have the short exact sequence

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

where p is the projection. Hence, E is an extension of R by M. It is trivial since ${\displaystyle r\mapsto (r,0)}$ is a section (note this section is a ring homomorphism since ${\displaystyle (1,0)}$ is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to ${\displaystyle R\oplus I}$ if ${\displaystyle I^{2}=0}$. Indeed, identifying ${\displaystyle R}$ as a subring of E using a section, we have ${\displaystyle (E,\phi )\simeq (R\oplus I,p)}$ via ${\displaystyle e\mapsto (\phi (e),e-\phi (e))}$.[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. 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 ${\displaystyle R/I}$-bimodule.

More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient ${\displaystyle R\to R_{\mathrm {red} }}$ 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
• The Wedderburn 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