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A New Construction of the Injective Hull

Published online by Cambridge University Press:  20 November 2018

Isidore Fleischer
Isidore Fleischer*
Affiliation:
Queens University
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The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 1 , April 1968 , pp. 19 - 21
Copyright
Copyright © Canadian Mathematical Society 1968

References

B Baer, R., Abelian groups that are direct summands of every containing abelian group, Bull. Am. Math. Soc. 46 (1940) 800-806.10.1090/S0002-9904-1940-07306-9Google Scholar
ES Eckmann, B. und Schopf, A., Über injektive Moduln, Arch. Math. 4 (1953) 75-78.10.1007/BF01899665Google Scholar
F Fleischer, I., Sur le problème d'application universelle de M. Bourbaki, C.R. Acad. Sci. Paris 254 (1962) 3161-3163.Google Scholar