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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Torsion-free abelian groups with prescribed finitely topologized endomorphism rings
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by Manfred Dugas and Rüdiger Göbel PDF
Proc. Amer. Math. Soc. 90 (1984), 519-527 Request permission

Abstract:

We will show that any complete Hausdorff ring $R$ which admits, as a basis of neighborhoods of 0, a family of right ideals $I$ with $R/I$ cotorsion-free can be realized as a topological endomorphism ring of some torsion-free abelian group with the finite topology. This theorem answers a question of A. L. S. Corner (1967) and can be used to provide examples in order to solve a problem (No. 72) in L. Fuchs’ book on abelian groups.
References
  • A. L. S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13 (1963), 687–710. MR 153743, DOI 10.1112/plms/s3-13.1.687
    • —, Endomorphism rings of torsion-free abelian groups, Proc. Internat. Conf. on the Theory of Groups, Australian National University, Canberra, 1965, Gordon & Breach, New York, 1967, pp. 59-69. —, A topological ring with the property that every non-trivial idempotent is the infinite orthogonal sum of idempotents, Abelian Group Theory (Proc. Honolulu, 1983), Lecture Notes in Math., vol. 1006, Springer, Berlin and New York (to appear).
  • Gabriella D’Este, A theorem on the endomorphism ring of reduced torsion-free abelian groups and some applications, Ann. Mat. Pura Appl. (4) 116 (1978), 381–392 (English, with Italian summary). MR 506986, DOI 10.1007/BF02413879
  • Gabriella D’Este, On topological rings which are endomorphism rings of reduced torsion-free abelian groups, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 127, 303–311. MR 625642, DOI 10.1093/qmath/32.3.303
  • Manfred Dugas and Rüdiger Göbel, Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), no. 2, 319–336. MR 670040, DOI 10.1112/plms/s3-45.2.319
  • Manfred Dugas and Rüdiger Göbel, On endomorphism rings of primary abelian groups, Math. Ann. 261 (1982), no. 3, 359–385. MR 679796, DOI 10.1007/BF01455456
  • Manfred Dugas and Rüdiger Göbel, Every cotorsion-free algebra is an endomorphism algebra, Math. Z. 181 (1982), no. 4, 451–470. MR 682667, DOI 10.1007/BF01182384
  • László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
  • Rüdiger Göbel, On stout and slender groups, J. Algebra 35 (1975), 39–55. MR 376879, DOI 10.1016/0021-8693(75)90034-4
  • Rüdiger Göbel and Burkhard Wald, Wachstumstypen und schlanke Gruppen, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 201–239 (German). MR 565607
    • R. Göbel and S. Shelah, Semi-rigid classes of cotorsion-free abelian groups, J. Algebra (submitted). R. Göbel, Überabzählbare abelsche Gruppen, Westdeutscher Verlag, Oldenburg (to appear).
  • Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561
  • Wolfgang Liebert, Endomorphism rings of abelian $p$-groups, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 239–258. MR 0242946
  • Klaus Meinel, Superdecomposable modules over Dedekind domains, Arch. Math. (Basel) 39 (1982), no. 1, 11–18. MR 674528, DOI 10.1007/BF01899239
  • Adalberto Orsatti, Anelli di endomorfismi di gruppi abeliani senza torsione, Symposia Mathematica, Vol. VIII (Convegno sulle Algebre Associative, INDAM, Rome, 1970) Academic Press, London, 1972, pp. 179–191 (Italian). MR 0338216
  • Adalberto Orsatti, A class of rings which are the endomorphism rings of some torsion-free abelian groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 23 (1969), 143–153. MR 242948
  • Luigi Salce and Federico Menegazzo, Abelian groups whose endomorphism ring is linearly compact, Rend. Sem. Mat. Univ. Padova 53 (1975), 315–325. MR 412303
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 519-527
  • MSC: Primary 20K20; Secondary 20K30
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0733399-8
  • MathSciNet review: 733399