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Vaught's Conjecture

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REFERENCES

  1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Oxford (1969).

    Google Scholar 

  2. J. T. Baldwin and R. N. McKenzie, “Counting models in universal Horn classes,” Algebra Universalis, 15, 359-384 (1982).

    Google Scholar 

  3. K. Bessenrodt, H. H. Brungs, and G. Törner, Right Chain Rings, Part 1, Schriftenreihe des Fachbereichs Math. Universität Duisburg, Vol.181 (1990).

  4. E. Bouscaren, “Martin's conjecture for ù-stable theories,” Israel J. Math., 49, Nos. 1-3, 15-25 (1984).

    Google Scholar 

  5. S. Buechler, “The classi.cation of small weakly minimal sets. I,” In: Classifcation Theory(J. T. Baldwin, Ed.), Chicago (1985), pp. 32-71.

  6. S. Buechler, “The classi.cation of small weakly minimal sets. III. Modules,” J. Symb. Logic, 53, No. 3, 975-979 (1988).

    Google Scholar 

  7. S. Buechler, “Vaught's conjecture for superstable theories of finite rank,” Ann. Pure Appl. Logic, to appear.

  8. S. Buechler, Vaught's conjecture for unidimensional theories, Preprint (1993).

  9. R. Camps and A. Facchini, “The Prüfer rings that are endomorphism rings of Artinian modules,” Commun. Algebra, 22, 3133-3157 (1994).

    Google Scholar 

  10. P. Eklof and I. Herzog, “Model theory of modules over a serial ring,” Ann. Pure Appl. Logic, 72, 145-176 (1995).

    Google Scholar 

  11. A. Facchini and G. Puninski, “∑-pure-injective modules over serial rings,” In: Abelian Groups and Modules(A. Facchini and C. Menini, Eds.), Kluwer Acad. Publ., Dordrecht (1995), pp. 145-162.

    Google Scholar 

  12. A. Facchini and G. Puninski, “Classical localizations in serial rings,” Commun. Algebra, 24, No. 11, 3537-3559 (1996).

    Google Scholar 

  13. C. Faith, Algebra: Rings, Modules, and Categories. II, Berlin, Springer (1976).

    Google Scholar 

  14. L. Fuchs and L. Salce, Modules over Valuation Domains, Lect. Notes Pure Appl. Math., Vol. 97 (1985).

  15. S. Garavaglia, “Decomposition of totally transcendental modules,” J. Symb. Logic, 45, No. 1, 155-164 (1980).

    Google Scholar 

  16. B. Hart, S. Starchenko, and M. Valeriote, “Vaught's conjecture for varieties,” Trans. Amer. Math. Soc., 342, No. 1, 173-196 (1994).

    Google Scholar 

  17. B. Hart and M. Valeriote, “A structure theorem for strongly Abelian varieties with few models,” J. Symb. Logic, 56, No. 3, 832-852 (1991).

    Google Scholar 

  18. I. Herzog, “Modules with few types,” J. Algebra, 149, 358-370 (1992).

    Google Scholar 

  19. I. Herzog and V. A. Puninskaya, “The model theory of divisible modules over a domain,” Fundam. Prikl. Mat., 2, No. 2, 563-594 (1996).

    Google Scholar 

  20. W. Hodges, Model Theory, Cambridge Univ., Cambridge (1993).

    Google Scholar 

  21. C. U. Jensen, “Arithmeticalri ngs,” Acta Math. Acad. Sci. Hungar., 17, 115-123 (1966).

    Google Scholar 

  22. D. Lascar, “Why some people are excited by Vaught's conjecture,” J. Symb. Logic, 50, No. 4, 973-982 (1985).

    Google Scholar 

  23. J. Loveys and P. Tanovic, “Countable models of trivial theories which admit finite coding,” J. Symb. Logic, 61, No. 4, 1279-1286 (1996).

    Google Scholar 

  24. L. F. Low and A. Pillay, “Superstable theories with few countable models,” Arch. Math. Logic, 31, 457-465 (1992).

    Google Scholar 

  25. L. Marcus, “The number of countable models of a theory of one unary operation,” Fundam. Math., 108, No. 3, 171-181 (1980).

    Google Scholar 

  26. L. L. Mayer, “Vaught's conjecture for O-minimalth eories,” J. Symb. Logic, 53, No. 1, 146-159 (1988).

    Google Scholar 

  27. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley, New York (1987).

    Google Scholar 

  28. A. Miller, “Vaught's conjecture for theories of one unary operation,” Fundam. Math., 111, No. 2, 135-141 (1981).

    Google Scholar 

  29. M. D. Morley, “The number of countable models,” J. Symb. Logic, 35, 14-18 (1970).

    Google Scholar 

  30. L. Newelski, “Vaught's conjecture for some meager groups,” Israel J. Math., 112, 271-299 (1999).

    Google Scholar 

  31. A. Pillay, An Introduction to Stability Theory, Oxford Logic Guides, Vol. 8, Clarendon Press, Oxford (1983).

    Google Scholar 

  32. A. Pillay, “Countable models of 1-based theories,” Arch. Math. Logic, 31, 163-169 (1992).

    Google Scholar 

  33. A. Pillay, On certain locally modular regular superstable groups, Preprint (1992).

  34. A. Pillay, Geometric Stability Theory, Oxford Logic Guides, Vol. 32, Clarendon Press, Oxford (1996).

    Google Scholar 

  35. M. Prest, Model Theory and Modules, London Math. Soc. Lect. Note Ser., Vol. 130, Cambridge Univ., Cambridge (1988).

    Google Scholar 

  36. M. Prest and V. A. Puninskaya, “Vaught's conjecture for modules over a commutative Prüfer ring,” Algebra Logika, 38, No. 4, 419-435 (1999).

    Google Scholar 

  37. M. Prest and V. Puninskaya, Modules with few types over some finite-dimensional algebras, Preprint, Manchester Univ., Manchester (2000).

  38. M. Prest and G. Puninski, “∑-pure-injective modules over a commutative Prüfer ring, Commun. Algebra, 27, 961-971 (1999).

    Google Scholar 

  39. V. Puninskaya, “Vaught's conjecture for modules over a Dedekind prime ring,” Bull. London Math. Soc. 31, No. 149, 129-135 (1999).

    Google Scholar 

  40. V. Puninskaya, “Vaught's conjecture for modules over a serial ring,” J. Symb. Logic, 65, No. 1, 155-163 (2000).

    Google Scholar 

  41. V. Puninskaya, “Modules with few types over a hereditary Noetherian prime ring,” J. Symb. Logic, to appear.

  42. V. Puninskaya, Modules with few types over a serial ring and over a commutative Prüfer ring, Preprint (2000).

  43. G. Puninski, M. Prest, and P. Rothmaler, “Rings described by various purities,” Commun. Algebra, 27, No. 5, 2127-2162 (1999).

    Google Scholar 

  44. G. E. Puninski and A. A. Tuganbaev, Rings and Modules[in Russian], Soyuz, Moscow (1998).

    Google Scholar 

  45. P. Rothmaler, Independence inU-rk 1 modules with few types, Preprint Christian-Albrechts-Universitat, Kiel(1987).

  46. L. H. Rowen, Ring Theory, Academic Press, New York (1991).

    Google Scholar 

  47. M. Rubin, “Theories of linear order,” Israel J. Math., 17, 392-443 (1974).

    Google Scholar 

  48. S. Shelah, L. A. Harrington, and M. Makkai, “A proof of Vaught's conjecture for ù-stable theories,” Israel J. Math., 49, 259-280 (1984).

    Google Scholar 

  49. J. Steel, “On Vaught's conjecture,” In: Cabal Seminar 76-77, (A. S. Kechris and Y. N. Moschovakis, Eds.), Lect. Notes Math., Vol. 689, Springer, Berlin (1978), pp. 193-208.

    Google Scholar 

  50. M. Valeriote, On decidable locally finite varieties, Ph.D. thesis, Univ. of California, Berkeley, California (1986).

    Google Scholar 

  51. R. L. Vaught, “Denumerable models of complete theories,” In: Proc. Symp. Found. Math. In.nitistic Methods, Pergamon Press, New York (1961), pp. 303-321.

    Google Scholar 

  52. C. M. Wagner, “Martin's conjecture for trees,” In: Abstr. Pap. Presented to Amer. Math. Soc., 2(1981), p. 528.

    Google Scholar 

  53. C. M. Wagner, “On Martin's conjecture,” Ann. Math. Logic, 22, 47-67 (1982).

    Google Scholar 

  54. R. B. Warfield, “Serial rings and finitely presented modules,” J. Algebra, 37, No. 3, 187-222 (1975).

    Google Scholar 

  55. M. Ziegler, “Model theory of modules,” Ann. Pure Appl. Logic, 26, No. 2, 149-213 (1984).

    Google Scholar 

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Puninskaya, V.A. Vaught's Conjecture. Journal of Mathematical Sciences 109, 1649–1668 (2002). https://doi.org/10.1023/A:1013985226489

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