In algebra, a ring extension of a ring R by an abelian group I is a pair (E, \( \ph \)i ) consisting of a ring E and a ring homomorphism \( \phi \) that fits into the short exact sequence of abelian groups:
\( {\displaystyle 0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.} \)
Note I is then a two-sided ideal of E. Given a commutative ring A, an A-extension 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 if \( \phi \) splits; i.e., \( \phi \) admits a section that is an algebra 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.
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
\( {\displaystyle (a,x)\cdot (b,y)=(ab,ay+bx).} \)
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. 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. One interesting feature of this construction is that the module M becomes an ideal of some new ring. In his "local rings", Nagata calls this process the principle of idealization.
References
E. Sernesi: Deformations of algebraic schemes
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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