Injective dimension and completeness
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- by M. Boratyński PDF
- Proc. Amer. Math. Soc. 35 (1972), 357-361 Request permission
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 ${S_\alpha }$.References
- Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR 74406, DOI 10.1017/S0027763000023291
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 357-361
- MSC: Primary 16A52
- DOI: https://doi.org/10.1090/S0002-9939-1972-0304428-2
- MathSciNet review: 0304428