Theorem 19.2.8. Let $\kappa $ be the cardinality of the set of ideals in $R$, and let $\alpha $ be an ordinal whose cofinality is greater than $\kappa $. Then $\mathbf{M}_\alpha (N)$ is an injective $R$-module, and $N \to \mathbf{M}_\alpha (N)$ is a functorial injective embedding.
Proof. By Baer's criterion Lemma 19.2.6, it suffices to show that if $\mathfrak {a} \subset R$ is an ideal, then any map $f : \mathfrak {a} \to \mathbf{M}_\alpha (N)$ extends to $R \to \mathbf{M}_\alpha (N)$. However, we know since $\alpha $ is a limit ordinal that
\[ \mathbf{M}_{\alpha }(N) = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } \mathbf{M}_{\beta }(N), \]
so by Proposition 19.2.5, we find that
\[ \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak {a}, \mathbf{M}_{\alpha }(N)) = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak a, \mathbf{M}_{\beta }(N)). \]
This means in particular that there is some $\beta ' < \alpha $ such that $f$ factors through the submodule $\mathbf{M}_{\beta '}(N)$, as
\[ f : \mathfrak {a} \to \mathbf{M}_{\beta '}(N) \to \mathbf{M}_{\alpha }(N). \]
However, by the fundamental property of the functor $\mathbf{M}$, see Lemma 19.2.7 part (3), we know that the map $\mathfrak {a} \to \mathbf{M}_{\beta '}(N)$ can be extended to
\[ R \to \mathbf{M}( \mathbf{M}_{\beta '}(N)) = \mathbf{M}_{\beta ' + 1}(N), \]
and the last object imbeds in $\mathbf{M}_{\alpha }(N)$ (as $\beta ' + 1 < \alpha $ since $\alpha $ is a limit ordinal). In particular, $f$ can be extended to $\mathbf{M}_{\alpha }(N)$. $\square$
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