Baer's Criterion

Theorem

Let $R$ be a ring with unity.

Let $M$ be a left $R$-module.

Then $M$ is injective if and only if the following condition holds:

For all left ideals $I$ of $R$ with inclusion map $\iota : I \to R$, and for all $R$-module homomorphisms $f : I \to M$, there exists an $R$-module homomorphism $\tilde f : R \to M$ such that:

$\tilde f \circ \iota = f$

Proof

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Source of Name

This entry was named for Reinhold Baer.