- References
In algebra, a ring extension of a ring R by an abelian group I is a pair (E, 
) consisting of a ring E and a ring homomorphism that fits into the exact sequence of abelian groups:
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 splits; i.e., 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 needs to be an isomorphism and 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
Note identifying (a, x) with a + εx, where ε squares to zero, and expanding (a + εx)(b + εy) out yield the above formula; in particular, we see E is a ring. We then have the exact sequence
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: [https://books.google.com/books/about/Deformations_of_algebraic_schemes.html?id=xkcpQo9tBN8C&hl=en Deformations of algebraic schemes]
- Paul S. Riebenfeld
- Paul S. Stull
- Paul S Stull
- Paul Stacey
- Paul Stackel
- Paul Staines
- Paul Stancliffe
- Paul-Stanislas LaRocque
- Paul-Stanislaus La Rocque
- Paul Stankowski
- Paul Stanley (director)
- Paul Stanley (disambiguation)
- Paul Stanley (legislator)
- Paul Stanley (politician)
- Paul Stanton (ice hockey)
- Paul Stapfer
- Paul Stapleton
- Paul starr
- Paul Starr
- Paul Starrett
- Paul Stastny
- Paul stastny
- Paul Staveley O'Duffy
- Paul Staveley O’Duffy
- Paul Stebbings