Ring extension

In algebra, a ring extension of a ring R by an abelian group I is a pair (E, ${\displaystyle \phi }$) consisting of a ring E and a ring homomorphism ${\displaystyle \phi }$ that fits into the exact sequence of abelian groups:

Note I is then an 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 ${\displaystyle \phi }$ splits; i.e., ${\displaystyle \phi }$ admits a section that is an algebra homomorphism.

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