Injective dimension and completeness

Abstract:

This paper contains the proofs of the two following theorems: (1) Let ${\{ {M_\alpha }\} _{\alpha < \gamma }}$ be a well-ordered decreasing system of submodules of the module M such that $M = {M_0}$. If M is strongly complete and strongly Hausdorff then \[ {\text {inj}} \dim M \leqq \sup \limits _{\alpha < \gamma } {\text {inj}} \dim {M_\alpha }/{M_{\alpha + 1}}.\] (2) Let R be a commutative ring having nonzero minimal idempotent ideals ${\{ {S_\alpha }\} _{\alpha < \gamma }}$ and let $S = \coprod \nolimits _{\alpha < \gamma } {{S_\alpha }}$. An R-module is injective if and only if M=Annih $S \oplus {M_0}$ where Annih S is injective and ${M_0}$ is strongly complete and Hausdorff in the topology introduced by annihilators of the direct sums of ${S_\alpha }$.