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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Injective dimension and completeness


Author: M. Boratyński
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
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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

$\displaystyle {\text{inj}}\,\dim \,M \leqq \mathop {\sup }\limits_{\alpha < \gamma } \,{\text{inj}}\,\dim \,{M_\alpha }/{M_{\alpha + 1}}.$

(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 }$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0304428-2
Article copyright: © Copyright 1972 American Mathematical Society