I tried to use Baer's criteria to disprove that $A$ is an injective $R$-module but that doesn't seem to work: I had hoped to find a short exact sequence that wouldn't split but the one that I got seems to be split
$$0 \to A \to R \to R/A \to 0.$$
Is $A=4\mathbb{Z}/12\mathbb{Z}$ an injective $R=\mathbb{Z}/12\mathbb{Z}$-module?
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homological-algebra
injective-module
1 Answers
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(My "answer" contained a terrible mistake: if $R$ is not a domain, then an injective $R$-module does not have to be divisible; for instance, $\mathbb{Z}/n\mathbb{Z}$ is injective over itself for any $n > 0$.)
You may indeed use Baer's criterion here. It's not about splitting short exact sequences though, it says that an $R$-module $M$ is injective if and only if for any ideal $\mathfrak{a} \subset R$ any $R$-linear map $\mathfrak{a} \to M$ extends to $R$.