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In commutative algebra, a ring extension is a ring homomorphism R → S {\displaystyle R\to S} of commutative rings, which makes S an R-algebra.
In this article, a ring extension of a ring R by an abelian group I is a pair of a ring E and a surjective ring homomorphism ϕ : E → R {\displaystyle \phi :E\to R} such that I is isomorphic to the kernel of ϕ . {\displaystyle \phi.} In other words,
is a short exact sequence of abelian groups.
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".
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