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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Problem 26 of L. Fuchs


Author: Chin Shui Hsü
Journal: Proc. Amer. Math. Soc. 42 (1974), 81-84
MSC: Primary 20K25
MathSciNet review: 0335660
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Abstract: This solves the following problem: Which Abelian groups are the inverse limits of Abelian groups, each of which is a finite direct sum of quasi-cyclic and bounded Abelian groups? (Here quasi-cyclic means isomorphic to some $ Z({p^\infty })$.) A necessary and sufficient condition for an Abelian group to be such is that it takes the form $ {A_r} \oplus \Pi_p\operatorname{Hom}_z({A_p},Z({p^\infty }))$ where $ {A_r}$ is complete and reduced, the $ {A_p}$ are torsion-free and the direct product is taken over the set of prime numbers.


References [Enhancements On Off] (What's this?)

  • [1] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673 (41 #333)
  • [2] C. U. Jensen, On the vanishing of \liminj⁽ⁱ⁾, J. Algebra 15 (1970), 151–166. MR 0260839 (41 #5460)
  • [3] A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Etudes Sci. Publ. Math. No. 11 (1961). MR 36 #177c.
  • [4] N. Bourbaki, Eléments de mathématique. XVII. Première partie: Les structures fondamentales de l’analyse. Livre I: Théorie des ensembles, Actualités Sci. Ind., no. 1212, Hermann & Cie, Paris, 1954 (French). Chapitre I: Description de la mathématique formelle. Chapitre II: Théorie des ensembles. MR 0065611 (16,454b)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0335660-1
PII: S 0002-9939(1974)0335660-1
Article copyright: © Copyright 1974 American Mathematical Society