Completely simple endomorphism rings of modules

• Autores: Victor Bovdi, Mohamed Salim, Mihail Ursul
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 19, Nº. 2, 2018, págs. 223-237
• Idioma: inglés
• DOI: 10.4995/agt.2018.7955
• Enlaces
  • Texto completo
• Resumen

  • It is proved that if Ap is a countable elementary abelian p-group, then: (i) The ring End (Ap) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (Ap)/I, where I is the ideal of End (Ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (Ap) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.

• Referencias bibliográficas
  • A. V. Arkhangelskii and V. I. Ponomarev, Osnovy obshchei topologii v zadachakh i uprazhneniyakh, Izdat. Nauka, Moscow, 1974.
  • V. I. Arnautov, S. T. Glavatsky and A. V. Mikhalev, Introduction to the theory of topological rings and modules, vol. 197 of Monographs and...
  • V. I. Arnautov and M. I. Ursul, Uniqueness of a linearly compact topology in rings, Mat. Issled. 53 (1979), 6-14, 221.
  • N. Bourbaki, Kommutativnaya algebra, Izdat. Mir, Moscow, 1971. Èlementy Matematiki, Vyp. XXVII, XXVIII, XXX, XXXI. [Foundations of Mathematics,...
  • N. Bourbaki, General topology. Chapters 1-4. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.
  • N. Bourbaki, Obshchaya topologiya. Izdat. Nauka, Moscow, 1975. Ispolzovanie veshchestvennykh chisel v obshchei topologii. Funktsionalnye prostranstva....
  • D. Dikranjan, Minimal topological rings, Serdica 8, no. 2 (1982), 149-165.
  • R. Engelking, General topology, vol. 6 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin, ii ed., 1989.
  • H. Freudenthal, Einige sätze über topologische gruppen, Ann. of Math. (2) 37, no. 1 (1936), 46-56. 1936.
  • E. D. Gaughan, Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 907-910. https://doi.org/10.1073/pnas.58.3.907
  • M. I. Graev, Theory of topological groups. I. Norms and metrics on groups. Complete groups. Free topological groups, Uspehi Matem. Nauk (N.S.)...
  • E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, vol. 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles...
  • M. Hochster and J. O. Kiltinen, Commutative rings with identity have ring topologies, Bull. Amer. Math. Soc. 76 (1970), 419-420. https://doi.org/10.1090/S0002-9904-1970-12495-8
  • A. Hulanicki, On locally compact topological groups of power of continuum, Fund. Math. 44 (1957), 156-158. https://doi.org/10.4064/fm-44-2-156-158
  • N. Jacobson, Totally disconnected locally compact rings, Amer. J. Math. 58, no. 2 (1936), 433-449. https://doi.org/10.2307/2371052
  • N. Jacobson, A note on topological fields, Amer. J. Math. 59, no. 4 (1937), 889-894. https://doi.org/10.2307/2371355
  • N. Jacobson, Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, American Mathematical Society, 190 Hope...
  • F. B. Jones, On the first countability axiom for locally compact Hausdorff spaces, Colloq. Math. 7 (1959), 33-34. https://doi.org/10.4064/cm-7-1-33-34
  • I. Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153-183. https://doi.org/10.2307/2371662
  • I. Kaplansky, Selected papers and other writings, Springer Collected Works in Mathe-matics, Springer, New York, 2013.
  • T. Y. Lam, A first course in noncommutative rings, vol. 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.
  • H. Leptin, Linear kompakte moduln und ringe, Math. Z. 62 (1955), 241-267. https://doi.org/10.1007/BF01180634
  • R. D. Mauldin, ed., The {S}cottish {B}ook, Birkhäuser/Springer, Cham, second ed., 2015.
  • A. F. Mutylin, Completely simple commutative topological rings, Mat. Zametki 5 (1969), 161-171. https://doi.org/10.1007/BF01098307
  • L. S. Pontryagin, Continuous groups, Nauka, Moscow, fourth ed., 1984.
  • L. Skornjakov, Einfache lokal bikompakte ringe, Math. Z. 87 (1965), 241-251. https://doi.org/10.1007/BF01109942
  • M. Ursul, Topological rings satisfying compactness conditions, vol. 549 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht,...
  • R. Ware and J. Zelmanowitz, Simple endomorphism rings, Amer. Math. Monthly 77 (1970), 987-989. https://doi.org/10.1080/00029890.1970.11992646
  • S. Warner, Topological fields, vol. 157 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1989. Notas de Matemática...
  • S. Warner, Topological rings, vol. 178 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1993.
  • D. Zelinsky, Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79-90. https://doi.org/10.2307/2372616