Completely simple endomorphism rings of modules

Victor Bovdi, Mohamed Salim, Mihail Ursul

Abstract

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.


Keywords

topological ring; endomorphism ring; Bohr topology; finite topology; locally compact ring.

Subject classification

16W80; 16N20; 16S50; 16N40.

Full Text:

PDF

References

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 Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1996.

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, No. XXVII, XXVIII, XXX, XXXI], Translated from the French by A. A. Belskii, Edited by E. S. Golod.

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. Svodka rezultatov. Slovar. [Application of real numbers in general topology. Functional spaces. Resumé of results. Vocabulary], Translated from the third French edition by S. N. Krackovskii, Edited by D. A. Raikov.

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. https://doi.org/10.2307/1968686

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.) 5, no. 2 (36) (1950), 3-56.

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, vol. 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin-New York, second ed., 1979. https://doi.org/10.1007/978-1-4419-8638-2

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 Street, Prov., R. I., 1956.

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. https://doi.org/10.1007/978-1-4684-0406-7

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, 2002. https://doi.org/10.1007/978-94-010-0249-3

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 [Mathematical Notes], 126.

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

Abstract Views

1082
Metrics Loading ...

Metrics powered by PLOS ALM




Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt