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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