Let be a mono in , and let be a map. We must extend to a map . Consider the poset whose elements are pairs where is an intermediate submodule between and and is an extension of , ordered by if contains and extends . By an application of Zorn's lemma, this poset has a maximal element, say . Suppose is not all of , and let be an element not in ; we show that extends to a map , contradiction.
The set is an ideal of , and we have a module map defined by . By hypothesis, we may extend to a module map . Writing a general element of as where , it may be shown that
is well-defined and extends , as desired.
Let be a Noetherian ring, and let be a collection of injective modules over . Then the direct sum is also injective.
By Baer’s criterion, it suffices to show that for any ideal of , a module map extends to a map . Since is Noetherian, is finitely generated as an -module, say by elements . Let be the projection, and put . Then for each , is nonzero for only finitely many summands. Taking all of these summands together over all , we see that factors through
for some finite . But a product of injectives is injective, hence extends to a map , which completes the proof.
Conversely, a result of Bass and Papp is that is Noetherian if direct sums of injective -modules are injective. See Lam, Theorem 3.46.