Ring extensions (Algebra)

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Information for Authority record

Name (Hebrew)
הרחבות חוגים (אלגברה)
Name (Latin)
Ring extensions (Algebra)
Name (Arabic)
امتدادات الحلقات (الجبر)
Other forms of name
Extensions of rings (Algebra)
See Also From tracing topical name
Rings (Algebra)
MARC
MARC

Other Identifiers

Wikidata: Q1166643
Library of congress: sh 85114119

Wikipedia description:

In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension. Precisely, a ring extension of a ring R by an abelian group I is a pair (E, ϕ {\displaystyle \phi } ) consisting of a ring E and a ring homomorphism ϕ {\displaystyle \phi } that fits into the short exact sequence of abelian groups: 0 → I → E → ϕ R → 0. {\displaystyle 0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.} 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 ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism (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.

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