Definition 2

Tag 0DPM, permanent. Cite it as [Bourbaki, Tag 0DPM].

One says that the A-module E is injective if, for every exact sequence of A-modules and homomorphisms

(19)

M' \xrightarrow{u} M \xrightarrow{v} M'',

the sequence of

\mathbf{Z}

-linear mappings

(20)

\operatorname{Hom}_A(M'', E) \xrightarrow{\operatorname{Hom}_A(v, 1)} \operatorname{Hom}_A(M, E) \xrightarrow{\operatorname{Hom}_A(u, 1)} \operatorname{Hom}_A(M', E)

is exact.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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