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Rings of continuous functions with values in a topological division ring

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Literature Cited

  1. V. T. Arnautov, M. I. Vodinchar, and A. V. Michalev,An Introduction to the Theory of Topological Rings and Modules [in Russian], St'iinitsa, Kishinev (1981).

    Google Scholar 

  2. A. V. Archangelsky, “Spaces of mappings and rings of continuous function,” In:Sovremennye Problemy Matematiki, Fundamental'nye Napravlenia, Vol. 51Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 81–171.

    Google Scholar 

  3. A. V. Archangelsky,Topological Spaces of Function, [in Russian], Moscow. State Univ., Moscow (1989).

    Google Scholar 

  4. A. V. Archangelsky and V. I. Ponomarev,Elements of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  5. G. E. Bredon,Theory of Sheaves [Russian translation], Nauka, Moscow (1988).

    Google Scholar 

  6. N. Bourbaki,General topology. Topological Groups. Numbers and Associated Spaces [Russian translation], Nauka, Moscow (1969).

    Google Scholar 

  7. N. Bourbaki,Commutative Algebra [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  8. E. M. Vechtomov, “On the module of all functions over the ring of continuous functions,”Mat. Zametki,28, 481–490 (1980).

    MATH  MathSciNet  Google Scholar 

  9. E. M. Vechtomov, “On projective and injective ideals of rings of continuous functions,” In:Abelian Groups and Modules [In Russian], Tomsk (1980), pp. 19–30.

  10. E. M. Vechtomov, “On ideals of rings of continuous functions,”Izv. Vuzov. Mat., No. 1, 3–10 (1981).

    MATH  MathSciNet  Google Scholar 

  11. E. M. Vechtomov, “On module of functions with bicompact supports over the ring of continuous functions,”Usp. Mat. Nauk,37, No. 4, 151–152 (1982).

    MATH  MathSciNet  Google Scholar 

  12. E. M. Vechtomov, “Distributive rings of continuous functions andF-spaces,”Mat. Zametki,34, 321–332 (1983).

    MATH  MathSciNet  Google Scholar 

  13. E. M. Vechtomov,On the theory of rings of continuous functions. I [in Russian], Tobol. Gos. Ped. Institute, Tobolsk (1985), Deposited at VINITI.

    Google Scholar 

  14. E. M. Vechtomov,On the theory of rings of continuous functions. II [in Russian], Tobol. Gos. Ped. Institute, Tobolsk (1985), Deposited at VINITI.

    Google Scholar 

  15. E. M. Vechtomov, “On rings of continuous functions with values in locally bicompact fields,” In:Abelian Groups and Modules [in Russian], Tomsk (1986), pp. 20–35.

  16. E. M. Vechtomov, “On reduced rings,” In:Semigroups and Partial Grouppoids [in Russian], LGPI. Leningrad (1987), pp. 3–14.

    Google Scholar 

  17. E. M. Vechtomov, “On rings of continuous functions with values in topological division rings,”Mat. Issled., No. 105, 45–52 (1988).

    MATH  MathSciNet  Google Scholar 

  18. E. M. Vechtomov, “On some properties of ideals in rings of continuous functions,” InTopological Spaces and Mappings [in Russian], Riga (1989), pp. 40–49.

  19. E. M. Vechtomov, “Pure ideals in rings and Bkouche theorem,” In:Abelian Groups and Modules [in Russian], Tomsk (1989), pp. 45–64.

  20. E. M. Vechtomov, “Problems of definability of topological spaces by algebraic systems of continuous functions,” In:Algebra. Topologia. Geometria. Vol. 26.Itogi Nauki i Tekhn. Vol. 26, All-Union Institute for Scientific and Technical Integration (VINITI), Akad Nauk SSSR, Moscow (1990), pp. 3–46.

    Google Scholar 

  21. E. M. Vechtomov, “Rings and modules of function,” In:Problems of Pure and Appl. Math., Abstr. of Reports [in Russian], Tartu (1990), pp. 108–111.

  22. E. M. Vechtomov, “On rings of continuous functions which are Bezout rings,” In:Abelian Groups and Modules [in Russian], Tomsk (1991), pp. 17–22.

  23. E. M. Vechtomov, “Rings of continuous functions. Algebraic aspects.” In:Itogi nauki i tekniki. Algebra. Topology [in Russian] (1991).

  24. E. M. Vechtomov,Rings of continuous functions on topological spaces. Selected themes [in Russian], Mosc. Ped. Gos. Univ., Moscow (1992).

    Google Scholar 

  25. E. M. Vechtomov, “On Gelfand-Kolmogorov theorem on maximal ideals in rings of continuos functions,”Usp. Mat. Nauk,47, No. 5, 171–172 (1992).

    MATH  MathSciNet  Google Scholar 

  26. E. M. Vechtomov,Functional Representations of Rings [in Russian], Mosk. Ped. Gos. Univ., Moscow (1993).

    Google Scholar 

  27. E. M. Vechtomov, “Rings of continuous functions and Gelfand theory,”Usp. Mat. Nauk,48, No. 1, 163–164 (1993).

    MATH  MathSciNet  Google Scholar 

  28. E. M. Vechtomov, “Rings of continuous functions and sheaves of rings,”Usp. Mat. Nauk,48, No. 5, 167–168 (1993).

    MATH  MathSciNet  Google Scholar 

  29. E. M. Vechtomov, “Rings and sheaves,” In:Sovremennaya Matematika i ee Prilozeniya. Tematicheskii obzory. 6. Topologiya-1, Itogi Nauki i Tekhn., All-Russian Tnstitute for Scientific and Technical Information (VINITI), Russian Akad. Nauk (in press).

  30. E. M. Vechtomov and A. G. Povyshev, “Idempotent and semiprime ideals in rings of continuous functions,”Latv. Univ. Zinatniskic Raksti Mat.,576, 63–74 (1992).

    MathSciNet  Google Scholar 

  31. I. M. Gelfand, “On normed rings.”Dokl. Akad. Nauk SSSR,23, No. 5, 430–432 (1939).

    MATH  Google Scholar 

  32. I. M. Gelfand and A. N. Kolmogorov, “On rings of continuous functions on topological spaces,”Dokl. Akad. Nauk. SSSR,22, No. 1, 11–15 (1939).

    Google Scholar 

  33. I. M. Gelfand, D. A. Raikov, and G. E. Shilov,Commutative Normed Rings [in Russian], Nauka, Moscow (1960).

    Google Scholar 

  34. R. Godement,Topologie Algebrique et Theorie des Faiseaux [Russian translation], Mir, Moscow (1961).

    Google Scholar 

  35. V. D. Golovin,Homologies of Analytic Sheaves and Duality theorems [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  36. A. V. Zarelua, “A method of theory of rings of functions in construction of bicompact extensions,” In:Contrib. Extens. Theory Topol. Struct. Proc. Sympos., Berlin (1969), pp 249–256.

  37. A. V. Zarelua, “A construction of strongly infinite compacts by the use of rings of continuous functions,”Dokl. Akad. Nauk SSSR,214, No. 2, 264–267 (1974).

    MATH  MathSciNet  Google Scholar 

  38. I. Lambek,Lectures in Rings and Modules, Chelsea Publishing, New York (1986).

    Google Scholar 

  39. M. A. Naimark,Normed rings [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  40. Edited by L. A. Skornjakov (eds.)General Algebra. Reference edition, Vol. 1, [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  41. L. S. Pontrjagin,Continuous Groups [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  42. A. A. Tuganbayev, “Distributive rings and modules,”Usp. Mat. Nauk,39, No. 1, 157–158 (1984).

    Google Scholar 

  43. M. I. Ursul, “On dimension theory of topological fields,”Izu. Akad. Nauk MSSR, Ser. Fiz.-tekh. Mat., No. 3, 47–48 (1987).

    MATH  MathSciNet  Google Scholar 

  44. A. Helemsky,Banach and Normed Algebras: General theory, Representations, Homologies [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  45. D. B. Sahmatov, “Cardinal invariants of topological fields,”Dokl. Akad. Nauk SSSR,271, No. 6, 1332–1336 (1983).

    MathSciNet  Google Scholar 

  46. R. Engelking,General topology [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  47. G. Bachman, E. Beckenstein, L. Narici, and S. Warner, “Rings of continuous function with values in a topological field,”Trans. Am. Math. Soc.,204, No. 4, 91–112 (1975).

    MathSciNet  Google Scholar 

  48. B. Banaschewski, “Uber nulldimensionale Raume,”Math. Nachr.,13, No. 3–4, 129–140 (1955).

    MATH  MathSciNet  Google Scholar 

  49. E. Beckenstein, L. Narici, and S. Suffel,Topological algebras, Oxford (1977).

  50. R. Bkouche, “Pureté, mollese et paracompactité,”C. R. Acad. Sci.,270, No. 25, A1653-A1655 (1970).

    MathSciNet  Google Scholar 

  51. F. Borceux, H. Simmons, and G. Van den Bossche, “A sheaf representation for modules with applications to Gelfand rings,”Proc. London Math. Soc.,48, No. 2, 230–246 (1984).

    MathSciNet  Google Scholar 

  52. J. G. Brookshear, “Projective ideals in rings of continuous functions,”Pacif. J. Math.,71, No. 2, 313–333 (1977).

    MATH  MathSciNet  Google Scholar 

  53. J. G. Brookshear, “On projective prime ideals inC(X),”Proc. Am. Math. Soc.,69, No. 1, 293–204 (1978).

    MathSciNet  Google Scholar 

  54. W. D. Burgess, K. A. Byrd, and R. Raphael, “Self-injective simple Pierce sheaves,”Arch. Math.,42, No. 4, 354–361 (1984).

    MathSciNet  Google Scholar 

  55. M. J. Canfell, “Uniqueness of generators of principal ideals in rings of continuous functions,”Proc. Am. Math. Soc.,26, No. 4, 571–573 (1990).

    MathSciNet  Google Scholar 

  56. S. U. Chase, “Direct product of modules,”Trans. Am. Math. Soc.,97, 457–473 (1960).

    MathSciNet  Google Scholar 

  57. P. R. Chernoff, R. A. Rasala, and W. C. Waterhouse, “The Stone-Weierstrass theorem for valuable fields,”Pacif. J. Math.,27, No. 2, 233–240 (1968).

    MathSciNet  Google Scholar 

  58. Kim-Peu Chew, “Structure of continuous functions. XI. Homomorphisms of some function rings,”Bull. Acad. Pol.Sci. Ser. Sci. Math. Astron. Phys.,19, No. 6, 485–489 (1971).

    MATH  MathSciNet  Google Scholar 

  59. G. Corach and F. D. Suárez, “Continuous selections and stable rank of Banach algebras,”Topol. Appl.,43, No. 3, 237–248 (1992).

    Google Scholar 

  60. E. Correll and M. Henriksen, “On rings of bounded continuous functions with values in a division ring,”Proc. Am. Math. Soc.,7, No. 2, 194–198 (1956).

    Google Scholar 

  61. B. J. Day, “Gelfand dualities over topological fields,”Proc. Austral. Math. Soc.,A32, No. 2, 171–177 (1982).

    MATH  MathSciNet  Google Scholar 

  62. J. M. Dominguiez, “NonarchimedeanC # (X),”Proc. Am. Math. Soc.,97, No. 3, 525–530 (1986).

    Google Scholar 

  63. R. Engelkeng and S. Mrówka, “OnE-compact spaces,”Bull. Acad. Pol. Sci. Ser. Math., Astron. Phys.,6, No. 7, 429–436 (1958).

    Google Scholar 

  64. R. L. Finney and J. Rotman, “Paracompactive of locally bicompact Hausdorff spaces,”Michigan Math. J.,17, No. 4, 359–361 (1970).

    MathSciNet  Google Scholar 

  65. L. Gillman, “Rings with Hausdorff structure space,”Fundam. Math.,45, No. 1, 11–16 (1957).

    MathSciNet  Google Scholar 

  66. L. Gillman and M. Henriksen, “Concerning rings of continuous functions,”Trans. Am. Math. Soc.,77, No. 2, 340–362 (1954).

    MathSciNet  Google Scholar 

  67. L. Gillman and M. Henriksen, “Rings of continuous functions in which every finitely ideal is principal,”Trans. Am. Math. Soc.,82, No. 2, 366–391 (1956).

    MathSciNet  Google Scholar 

  68. L. Gillman, M. Henriksen, and M. Jerison, “On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions,”Proc. Am. Math. Soc.,5, No. 3, 447–455 (1954).

    MathSciNet  Google Scholar 

  69. L. Gillman and M. Jerison,Rings of continuous functions, Van Nostrand (1960).

  70. L. Gillman and C. W. Kohls, “Convex and pseudoprime ideals in rings of continuous functions,”Math. Z.,72, No. 5, 399–409 (1960).

    MathSciNet  Google Scholar 

  71. J. K. Goldhaber and E. S. Wolk, “Maximal ideals in rings of bounded continuous functions,”Duke Math. J.,21, No. 3, 565–569 (1954).

    MathSciNet  Google Scholar 

  72. M. Henriksen, “Involved problem on algebraic aspects ofC|X|,” [93], 195–202.

  73. E. Hewitt, “Rings of real-valued continuous functions. I,”Trans. Am. Math. Soc.,64, No. 1, 45–99 (1948).

    MATH  MathSciNet  Google Scholar 

  74. K. H. Hofmann, “Representations of algebras by continuous sections,”Bull. Am. Math. Soc.,78, No. 3, 291–373 (1972).

    MATH  Google Scholar 

  75. Pao-Sheng Hsu, “An application of compactifications: some theorems on maximal ideals,”Ann. Mat. Pure Appl.,112, 107–118 (1977).

    MATH  Google Scholar 

  76. C. Huijsmans and B. Pagter, “Ideal theory inf-algebras,”Trans. Am. Math. Soc.,269, No. 1, 225–245 (1982).

    Google Scholar 

  77. J. Isbell, “Zero-dimensional spaces,”Tohoku Math. J.,7, No. 1–2, 1–8 (1955).

    MATH  MathSciNet  Google Scholar 

  78. I. Kaplansky, “Topological rings,”Am. J. Math.,69, 153–183 (1947).

    MATH  MathSciNet  Google Scholar 

  79. I. Kaplansky, “Topological methods in valuation theory,”Duke Math. J.,14, No. 3, 527–541 (1947).

    MathSciNet  Google Scholar 

  80. K. Koh, “On a representation of a strongly harmonic ring by sheaves,”Pacif. J. Math.,41, No. 2, 459–468 (1972).

    MATH  MathSciNet  Google Scholar 

  81. J. Lambek, “On the representation of modules by sheaves of factor modules,”Can. Math. Bull. 14, No. 3, 359–368 (1971).

    MATH  MathSciNet  Google Scholar 

  82. A. Mallios,Topological algebras. Selected topics, North-Holland, Amsterdam (1986).

    Google Scholar 

  83. G. Marco, “Projectivity of pure ideals,”Rend. Semin. Math. Univ. Padova,69, 289–304 (1983).

    MATH  Google Scholar 

  84. G. Macro and A. Orsatti, “Commutative rings in which every prime ideal is contained in a unique maximal ideal,”Proc. Am. Math. Soc.,30, No. 3, 459–466 (1971).

    Google Scholar 

  85. G. Marco and M. Richter, “Rings of continuous functions with values in a nonarchimedean ordered field,”Rend. Semin. Mat. Univ. Padova,45, 327–336 (1971).

    Google Scholar 

  86. G. Marco and R. G. Wilson, “Rings of continuous functions with values in an archimedean ordered field,”Rent. Semin. Mat. Univ. Padova,44, 263–272 (1970).

    Google Scholar 

  87. S. Mrówka, “Further results onE-compact spaces. I,”Acta Math. Acad. Sci. Hungary,120, 161–185 (1968).

    MATH  Google Scholar 

  88. S. Mrówka, “Structures of continuous functions. I,”Acta Math. Acad. Sci. Hungary,21, 239–259 (1970).

    MATH  Google Scholar 

  89. C. J. Mulvey, “Compact ringed spaces,”J. Algebra,52, No. 2, 411–436 (1878).

    MathSciNet  Google Scholar 

  90. C. J. Mulvey, “A generalization of Gelfand duality,”J. Algebra,56, No. 2, 499–505 (1979).

    MATH  MathSciNet  Google Scholar 

  91. C. J. Mulvey, “Representations of rings and modules,”Lect. Notes Math.,753, 542–585 (1979).

    MATH  MathSciNet  Google Scholar 

  92. r. S. Pierce, “Modules over commutative regular rings,”Mem. Am. Math. Soc.,70, 1–112 (1967).

    MathSciNet  Google Scholar 

  93. R. S. Pierce,Rings of cantinuous functions, Marcel Dekker, New York (1985).

    Google Scholar 

  94. A. Rooij and C. V. Van,Non-Archimedean functional analysis, Marcel Dekker, New York (1978).

    Google Scholar 

  95. D. Rudd, “On two sum theorems for ideals ofC(X),”Michigan Math. J.,17, No. 2, 139–141 (1970).

    MATH  MathSciNet  Google Scholar 

  96. N. Shilkret, “Non-Archimedean Gelfand theory,”Pacif. J. Math.,32, No. 2, 541–550 (1970).

    MATH  MathSciNet  Google Scholar 

  97. H. Simmons, “Sheaf representations of strongly harmonic rings”,Proc. Roy Soc. Edinburgh,A99, No. 3–4, 249–268 (1985).

    MATH  MathSciNet  Google Scholar 

  98. M. Stone, “Applications of the theory of boolean rings to general topology,”Trans. Am. Math. Soc.,41, No. 3, 375–481 (1937).

    MATH  Google Scholar 

  99. Shu-Hao Sun, “Noncommutative rings in which every prime ideal is contained in a unique maximal ideal,”J. Pure Appl. Algebra,76, No. 2, 179–192 (1991).

    MATH  MathSciNet  Google Scholar 

  100. S. Warner, “Characters of Cartesian products of algebras,”Can. J. Math.,11, No. 1, 70–79 (1959).

    MATH  MathSciNet  Google Scholar 

  101. S. Warner,Topological fields, North-Holland, Amsterdam (1989).

    Google Scholar 

  102. W. Wieslaw,Topological fields, Wroclav (1982).

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i ee Prilozheniya. Tematicheskiye Obzory. Vol. 14, Topologiya-2, 1994

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Vechtomov, E.M. Rings of continuous functions with values in a topological division ring. J Math Sci 78, 702–753 (1996). https://doi.org/10.1007/BF02363066

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