Every Module Is an Inverse Limit of Injectives

It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised. The flat modules over a ring are precisely the direct limits of projective modules [11] [6] [10, Theorem 2.4.34]. Which modules are, dually, inverse limits of injectives? I sketched the answer in [1], but in view of the limited distribution of that item, it seems worthwhile to make the result more widely available. The construction from [1] is Theorem 2 below; the connecting maps there are inclusions. In Theorem 4, we shall see that the connecting maps can, alternatively, be taken to be onto if R is Noetherian on the appropriate side. In §2 we ask some questions, in §3 we take some steps toward answering one of them, and in §4 we note what the proofs of our results tell us when applied to not necessarily injective modules. Throughout, “ring” means associative ring with unit, and modules are unital. I am indebted to Pace Nielsen for pointing out the need to assume κ regular in Lemma 1, and to the referee for some useful suggestions.