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Properties of endomorphism rings of Abelian groups, I

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REFERENCES

  1. U. Albrecht, “Endomorphism rings and A-projective torsion-free Abelian groups,” Lect. Notes Math., 1006, 209-227 (1983).

    Google Scholar 

  2. U. Albrecht, “A-projective groups of large cardinality,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 233-242.

    Google Scholar 

  3. U. Albrecht, “A note on locally A-projective groups,” Pacific J. Math., 120, No. 1, 1-17 (1985).

    Google Scholar 

  4. U. Albrecht, “Chain conditions in endomorphism rings,” Rocky Mountain J. Math., 15, No. 1, 91-106 (1985).

    Google Scholar 

  5. U. Albrecht, “Baer's lemma and Fuchs's problem 84a,” Trans. Amer. Math. Soc., 293, No. 2, 565-582 (1986).

    Google Scholar 

  6. U. Albrecht, “Abelian groups with self-injective endomorphism rings,” Comm. Algebra, 15, No. 12, 2451-2471 (1987).

    Google Scholar 

  7. U. Albrecht, “Faithful Abelian groups of infinite rank,” Proc. Amer. Math. Soc., 103, No. 1, 21-26 (1988).

    Google Scholar 

  8. U. Albrecht, “The structure of generalized rank one groups,” Houston J. Math., 14, No. 3, 305-317 (1988).

    Google Scholar 

  9. U. Albrecht, “The structure of generalized rank one groups,” Houston J. Math., 14, No. 3, 305-317 (1988).

    Google Scholar 

  10. U. Albrecht, “A-reflexive Abelian groups,” Houston J. Math. 15, No. 4, 459-480 (1989).

    Google Scholar 

  11. U. Albrecht, “Endomorphism rings and a generalization of torsion-freeness and purity,” Comm. Algebra, 17, No. 5, 1101-1135 (1989).

    Google Scholar 

  12. U. Albrecht, “Abelian groups A such that the category of A-solvable groups is preabelian,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 117-131.

    Google Scholar 

  13. U. Albrecht, “Endomorphism rings of faithfully flat Abelian groups,” Results Math., 17, No. 3-4, 179-201 (1990).

    Google Scholar 

  14. U. Albrecht, “Locally A-projective Abelian groups and generalizations,” Pacific J. Math. 141, No. 2, 209-228 (1990).

    Google Scholar 

  15. U. Albrecht, “Extension functors on the category of A-solvable Abelian groups,” Czechosl. Math. J., 41, No. 4, 685-694 (1991).

    Google Scholar 

  16. U. Albrecht, “Endomorphism rings, tensor products and Fuchs' problem 47,” In: Abelian Groups and Noncommutative Rings, Amer. Math. Soc., Providence, RI (1992), pp. 17-31.

    Google Scholar 

  17. U. Albrecht, “Quasi-decomposition of Abelian groups and Baer's lemma,” Rocky Mountain J. Math., 22, No. 4, 1227-1241 (1992).

    Google Scholar 

  18. U. Albrecht, “Abelian groups with semi-simple Artinian quasi-endomorphism rings,” Rocky Mountain J. Math., 24, No. 3, 853-866 (1994).

    Google Scholar 

  19. U. Albrecht, “The construction of A-solvable Abelian groups,” Czechosl. Math. J., 44, No. 3, 413-430 (1994).

    Google Scholar 

  20. U. Albrecht, “Mixed Abelian groups with Artinian quasi-endomorphism ring,” Comm. Algebra 25, No. 11, 3497-3511 (1997).

    Google Scholar 

  21. U. Albrecht and T. Faticoni, “Abelian groups flat as modules over their endomorphism ring,” Comm. Algebra, 21, No. 10, 3403-3423 (1993).

    Google Scholar 

  22. U. Albrecht, T. Faticoni, and H. Goeters, “A note on coflat Abelian groups,” Czechosl. Math. J., 44, No. 3, 431-442 (1994).

    Google Scholar 

  23. U. Albrecht and H. Goeters, “Flatness and the ring of quasi-endomorphisms,” Quaest. Math., 19, No. 1-2, 379-396 (1996).

    Google Scholar 

  24. U. Albrecht, H. Goeters, T. Faticoni, and W. Wickless, “Subalgebras of rational matrix algebras,” Acta Math. Hungar., 74, No. 1-2, 1-6 (1997).

    Google Scholar 

  25. U. Albrecht, H. Goeters, and W. Wickless, “The flat dimension of mixed Abelian groups as Emodules,” Rocky Mountain J. Math., 25, No. 2, 569-590 (1995).

    Google Scholar 

  26. U. Albrecht and J. Hausen, “Modules with the quasi-summand intersection property,” Bull. Austral. Math. Soc. 44, No. 2, 189-201 (1991).

    Google Scholar 

  27. U. Albrecht and J. Hausen, “Mixed Abelian groups with the summand intersection property,” In: Abelian Groups and Modules (Colorado Springs, CO, 1995), Dekker, New York (1996), pp. 123-132.

    Google Scholar 

  28. H. W. K. Angad-Gaur, “The homological dimension of a torsion-free Abelian group of finite rank as a module over its ring of endomorphisms,” Rend. Sem. Mat. Univ. Padova., 57 (1977); 299-309 (1978).

    Google Scholar 

  29. D. Arnold, “Strongly homogeneous torsion free Abelian groups of finite rank,” Proc. Amer. Math. Soc., 56, 67-72 (1976).

    Google Scholar 

  30. D. Arnold, “Endomorphism rings and subgroups of finite rank torsion-free Abelian groups,” Rocky Mountain J. Math., 12, No. 2, 241-256 (1982).

    Google Scholar 

  31. D. Arnold, “Finite rank torsion free Abelian groups and rings,” Lect. Notes Math., 931, 1-191 (1982).

    Google Scholar 

  32. D. Arnold, “Butler groups, representations of finite posets, and modules over multiple pullback rings,” In: Methods in Module Theory (Colorado Springs, CO, 1991), Dekker, New York (1993), pp. 1-18.

    Google Scholar 

  33. D. Arnold, “Abelian groups flat as modules over their endomorphism rings,” Preprint.

  34. D. Arnold and J. Hausen, “A characterization of modules with the summand intersection property,” Comm. Algebra, 18, No. 2, 519-528 (1990).

    Google Scholar 

  35. D. Arnold and E. L. Lady, “Endomorphism rings and direct sums of torsion free Abelian groups,” Trans. Amer. Math. Soc., 211, 225-237 (1975).

    Google Scholar 

  36. D. Arnold and C. E. Murley, “Abelian groups, A, such that Hom(A,?) preserves direct sums of copies of A,” Pacific J. Math., 56, No. 1, 7-20 (1975).

    Google Scholar 

  37. D. Arnold, R. S. Pierce, J. D. Reid, C. Vinsonhaler, and W. Wickless, “Torsion-free Abelian groups of finite rank projective as modules over their endomorphism rings,” J. Algebra 71, No. 1, 1-10 (1981).

    Google Scholar 

  38. D. Arnold, B. O'Brien, and J. D. Reid, “Quasipure injective and projective torsion-free abelian groups of finite rank,” Proc. London Math. Soc. (3), 38, No. 3, 532-544 (1979).

    Google Scholar 

  39. D. Arnold and C. Vinsonhaler, “Endomorphism rings of Butler groups,” J. Austral. Math. Soc. Ser. A, 42, No. 3, 322-329 (1987).

    Google Scholar 

  40. D. Arnold and C. Vinsonhaler, “Quasi-endomorphism rings for a class of Butler groups,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 85-89.

    Google Scholar 

  41. D. Arnold and C. Vinsonhaler, “Isomorphism invariants for Abelian groups,” Trans. Amer. Math. Soc. 330, No. 2, 711-724 (1992).

    Google Scholar 

  42. D. Arnold, C. Vinsonhaler, and W. J. Wickless, “Quasi-pure projective and injective torsion free abelian groups of rank 2,” Rocky Mountain J. Math., 6, No. 1, 61-70 (1976).

    Google Scholar 

  43. M. N. Arshinov, “The projective dimension of torsion-free Abelian groups over their endomorphism ring,” Mat. Zametki, 7, 117-124 (1970).

    Google Scholar 

  44. E. Artin, Geometric Algebra, Interscience Publishers, New York (1957).

    Google Scholar 

  45. M.-A. Avino Diaz, “Degree of nilpotency of the principal group of congruences modulo the Jacobson radical of the endomorphism ring of a finite Abelian p-group of type [p m 1, p m 2,⋯, pm r ], where mi-1 > m i + 1 for i = 2, 3,⋯, r,” Cienc. Mat. (Havana), 5, No. 3, 43-51 (1984).

    Google Scholar 

  46. M.-A. Avino Diaz and S. Rodriguez Maribona, “The nilpotency class of the maximum normal psubgroup of the group of automorphisms of a finite Abelian p-group of type (p m 1, p m 2,⋯, p m r ), with m i-1 = m i +1, i = 2,⋯, r,” Cienc. Mat. (Havana), 6, No. 2, 51-55 (1985).

    Google Scholar 

  47. R. Baer, “Abelian groups without elements of finite order,” Duke Math. J., 3, 68-122 (1937).

    Google Scholar 

  48. R. Baer, “A unified theory of projective spaces and finite Abelian groups,” Trans. Amer. Math. Soc., 52, 283-343 (1942).

    Google Scholar 

  49. R. Baer, “Automorphism rings of primary Abelian operator groups,” Ann. Math., 44, 192-227 (1943).

    Google Scholar 

  50. R. Baer, Linear Algebra and Projective Geometry, Academic Press, New York (1952).

    Google Scholar 

  51. H. Bass, Algebraic K-Theory, Benjamin, New York (1968).

    Google Scholar 

  52. S. Bazzoni and C. Metelli, “On Abelian torsion-free separable groups and their endomorphism rings,” In: 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. 259-285.

    Google Scholar 

  53. R. A. Beaumont and R. S. Pierce, “Torsion-free rings,” Ill. J. Math., 5, 61-98 (1961).

    Google Scholar 

  54. R. A. Beaumont and R. S. Pierce, “Subrings of algebraic number fields,” Acta Sci. Math. Szeged, 22, 202-216 (1961).

    Google Scholar 

  55. R. Behler and R. Göbel, “Abelian p-groups of arbitrary length and their endomorphism rings,” Note Mat., 11, 7-20 (1991).

    Google Scholar 

  56. R. Behler, R. Göbel, and R. Mines, “Endomorphism rings of p-groups having length cofinal with ?,” In: Abelian Groups and Noncommutative Rings, Amer. Math. Soc., Providence, RI (1992), pp. 33-48.

    Google Scholar 

  57. K. I. Beidar, W. S. Martindale, III, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, New York, (1996).

    Google Scholar 

  58. K. I. Beidar, A. V. Mikhalev, and G. E. Puninski, “Logical aspects of the theory of rings and modules,” Fundament. Prikl. Mat., 1, No. 1, 1-62 (1995).

    Google Scholar 

  59. I. Kh. Bekker and S. F. Kozhukhov, Automorphisms of Torsion-Free Abelian Groups, Tomsk. Gos. Univ., Tomsk (1988).

    Google Scholar 

  60. I. Kh. Bekker and S. K. Rososhek, “Polynomial splittability and weakly rigid systems of torsion-free Abelian groups,” In: Abelian Groups and Modules, Tomsk. Gos. Univ., Tomsk >(1979), pp. 3-13.

    Google Scholar 

  61. K. Benabdallah, Groupes Abeliens sans Torsion, Presses de l'Université de Montreal, Montreal (1981).

    Google Scholar 

  62. K. Benabdallah and A. Birtz, “p-Pure envelopes of pairs in torsion free Abelian groups,” Comment. Math. Univ. St. Paul., 28, No. 1, 107-114 (1980).

    Google Scholar 

  63. K. Benabdallah and A. Birtz, “Sur une famille de groupes abeliens super-decomposables,” Canad. Math. Bull., 24, No. 2, 213-218 (1981).

    Google Scholar 

  64. G. Birkenmeier and H. Heatherly, “Rings whose additive endomorphisms are ring endomorphisms,” Bull. Austral. Math. Soc., 42, No. 1, 145-152 (1990).

    Google Scholar 

  65. I. V. Bobylev, “Projective dimension of an Abelian group over its endomorphism ring,” Usp. Mat. Nauk, 28, No. 2, 229-230 (1973).

    Google Scholar 

  66. I. V. Bobylev, “Rings over which every module is endoprojective,” Usp. Mat. Nauk, 29, No. 3, 182 (1974).

    Google Scholar 

  67. I. V. Bobylev, “Endoprojective dimension of modules,” Sib. Mat. Zh., 16, No. 4, 663-683 (1975).

    Google Scholar 

  68. I. V. Bobylev, “Uniserial rings,” Mat. Zametki, 38, No. 1, 35-43 (1985).

    Google Scholar 

  69. J. D. Botha and P. J. Grabe, “On torsion-free Abelian groups whose endomorphism rings are principal ideal domains,” Comm. Algebra, 11, No. 12, 1343-1354 (1983).

    Google Scholar 

  70. N. Bourbaki, Algébre Commutative, Hermann, (1961, 1964, 1965).

  71. H. Bowman and K. M. Rangaswamy, “Torsion-free separable Abelian groups quasi-projective over their endomorphism rings,” Houston J. Math., 11, No. 4, 447-453 (1985).

    Google Scholar 

  72. R. A. Bowshell and P. Schultz, “Unital rings whose additive endomorphisms commute,” Math. Ann., 228, No. 3, 197-214 (1977).

    Google Scholar 

  73. S. Brenner and M. C. R. Butler, “Endomorphism rings of vector spaces and torsion free Abelian groups,” J. London Math. Soc., 40, 183-187 (1965).

    Google Scholar 

  74. M. C. R. Butler, “On locally free torsion-free rings of finite rank,” J. London Math. Soc., 43, 297-300 (1968).

    Google Scholar 

  75. H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press (1956).

  76. B. Charles, “Le centre de l'anneau des endomorphismes d'un groupe abelien primaire,” C. R. Acad. Sci., 236, 1122-1123 (1953).

    Google Scholar 

  77. B. Charles, “Sous-groupes fonctoriels et topologies,” In: Studies on Abelian Groups (Symposium, Montpellier, 1967), Springer, Berlin (1968), pp. 75-92.

    Google Scholar 

  78. A. R. Chekhlov, “Some classes of Abelian groups,” In: Abelian Groups and Modules, Tomsk. Gos. Univ., Tomsk, 137-152 (1984).

    Google Scholar 

  79. A. R. Chekhlov, “Some classes of torsion-free Abelian groups that are close to quasipure injective groups,” Izv. Vuzov. Mat., No. 8, 82-83 (1985).

  80. A. R. Chekhlov, “Torsion-free Abelian groups close to quasipure injective,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1985), pp. 117-127.

    Google Scholar 

  81. A. R. Chekhlov, “Quasipure injective torsion-free Abelian groups,” Izv. Vuzov. Mat., No. 6, 80-83 (1988).

    Google Scholar 

  82. A. R. Chekhlov, “Torsion-free Abelian CS-groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1988), pp. 131-147.

    Google Scholar 

  83. A. R. Chekhlov, “Quasipure injective torsion-free Abelian groups,” Mat. Zametki, 46, No. 3, 93-99 (1989).

    Google Scholar 

  84. A. R. Chekhlov, “Cohesive quasipure injective Abelian groups,” Izv. Vuzov. Mat., No. 10, 84-87 (1989).

  85. A. R. Chekhlov, “On quasipure injective torsion-free Abelian groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1989), pp. 139-153.

    Google Scholar 

  86. A. R. Chekhlov, “Abelian torsion-free CS-groups,” Izv. Vuzov. Mat., No. 3, 84-87 (1990).

    Google Scholar 

  87. . A. R. Chekhlov, “Direct products and direct sums of torsion-free Abelian QCPI-groups,” Izv. Vuzov. Mat., No. 4, 58-67 (1990).

  88. A. R. Chekhlov, “Torsion-free Abelian groups of finite p-rank with complemented closed pure subgroups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1991), pp. 157-178.

    Google Scholar 

  89. A. V. Cherednikova, “Quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 3,” Mat. Zametki, 63, No. 5, 763-773 (1998).

    Google Scholar 

  90. J. M. Chung, “On regular groups over their endomorphism rings,” Commun. Korean Math. Soc. 11, No. 2, 311-314 (1996).

    Google Scholar 

  91. R. R. Colby and E. A. Rutter, “Generalizations of QF-3 algebras,” Trans. Amer. Math. Soc., 153, 371-386 (1971).

    Google Scholar 

  92. E. F. Cornelius, “Characterization of a class of torsion free groups in terms of endomorphisms,” Pacific J. Math. 79, No. 2, 341-355 (1978).

    Google Scholar 

  93. A. L. S. Corner, “Every countable reduced torsion-free ring is an endomorphism ring,” Proc. London Math. Soc. (3), 13 687-710 (1963).

    Google Scholar 

  94. A. L. S. Corner, “On a conjecture of Pierce concerning direct decompositions of Abelian groups,” In: Proc. Colloq. Abelian Groups (Tihany, 1963), Akademiai Kiado, Budapest (1964), pp. 43-48.

    Google Scholar 

  95. A. L. S. Corner, “Endomorphism rings of torsion-free Abelian groups,” In: Proc. Internat. Conf. Theory of Groups, Canberra (1965), pp. 59-69.

  96. A. L. S. Corner, “On endomorphism rings of primary Abelian groups,” Quart. J. Math. Oxford Ser. (2), 20, 277-296 (1969).

    Google Scholar 

  97. A. L. S. Corner, “On endomorphism rings of primary Abelian groups. II” Quart. J. Math. Oxford Ser. (2), 27, No. 105, 5-13 (1976).

    Google Scholar 

  98. A. L. S. Corner, “On the existence of very decomposable Abelian groups,” Lect. Notes Math., 1006, 354-357 (1983).

    Google Scholar 

  99. A. L. S. Corner and R. Göbel, “Prescribing endomorphism algebras, a unified treatment,” Proc. London Math. Soc. (3), 50, No. 3, 447-479 (1985).

    Google Scholar 

  100. A. L. S. Corner and R. Göbel, “Subgroups of the Baer-Specker group with prescribed endomorphism ring,” In: Abelian Groups, Module Theory, and Topology (Padua 1997), Dekker, New York (1998), pp. 113-173.

    Google Scholar 

  101. E.-G. Dawley, “An ote on Abelian p-groups and their endomorphism rings,” In: Black Mathematicians and Their Works, Dorrance, Ardmore, Pa. (1980), pp. 137-138.

    Google Scholar 

  102. G. D'Este, “Ath eorem on the endomorphism ring of reduced torsion-free Abelian groups and some applications,” Ann. Mat. Pura Appl. (4), 116, 381-392 (1978).

    Google Scholar 

  103. G. D'Este, “Abelian groups with anti-isomorphic endomorphism rings,” Rend. Sem. Mat. Univ. Padova., 60 (1978), 55-75 (1979).

    Google Scholar 

  104. G. D'Este, “On topological rings which are endomorphism rings of reduced torsion-free Abelian groups,” Quart. J. Math. Oxford Ser. (2), 32, No. 127, 303-311 (1981).

    Google Scholar 

  105. J. Dieudonne, La Gëomëtrie des Groups Classiques, Springer-Verlag, Berlin (1971).

    Google Scholar 

  106. N. T. Djao, “Endomorphism rings of Abelian groups as isomorphic restrictions of full endomorphism rings,” Period. Math. Hungar., 24, No. 3, 129-133 (1992).

    Google Scholar 

  107. N. T. Djao, “Endomorphism rings of Abelian groups as isomorphic restrictions of full endomorphism rings. II,” Acta Math. Hungar., 59, No. 3-4, 253-262 (1992).

    Google Scholar 

  108. Yu. B. Dobrusin, “Quasi-pure injective groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1979), pp. 45-63.

    Google Scholar 

  109. Yu. B. Dobrusin, “Torsion-free quasi-pure injective Abelian groups,” In: Abelian Groups and Modules, Tomsk. Gos. Univ., Tomsk (1980), pp. 45-69.

    Google Scholar 

  110. Yu. B. Dobrusin, “Extensions of partial endomorphisms of torsion-free Abelian groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1985), pp. 36-53.

    Google Scholar 

  111. Yu. B. Dobrusin, “Extensions of partial endomorphisms of torsion-free Abelian groups, II” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1985), pp. 31-41.

    Google Scholar 

  112. A. J. Douglas and H. K. Farahat, “The homological dimension of an Abelian group as a module over its ring of endomorphisms,” Monatsh. Math., 69, 294-305 (1965).

    Google Scholar 

  113. A. J. Douglas and H. K. Farahat, “The homological dimension of an Abelian group as a module over its ring of endomorphisms. II,” Monatsh. Math., 76, 109-111 (1972).

    Google Scholar 

  114. A. J. Douglas and H. K. Farahat, “The homological dimension of an Abelian group as a module over its ring of endomorphisms. III,” Monatsh. Math., 80, No. 1, 37-44 (1975).

    Google Scholar 

  115. D. W. Dubois, “Cohesive groups and p-adic integers,” Publ. Math. Debrecen, 12, 51-58 (1965).

    Google Scholar 

  116. M. Dugas, “Fast freie abelsche Gruppen mit Endomorphismenring Z,” J. Algebra, 71, No. 2, 314-321 (1981).

    Google Scholar 

  117. M. Dugas, “On the existence of large mixed modules,” Lect. Notes Math., 1006, 412-424 (1983).

    Google Scholar 

  118. M. Dugas, “On the Jacobson radical of some endomorphism rings,” Proc. Amer. Math. Soc., 102, No. 4, 823-826 (1988).

    Google Scholar 

  119. M. Dugas and T. Faticoni, “Cotorsion-free Abelian groups: Cotorsion as modules over their endomorphism rings,” In: Abelian Groups (Curacao, 1991), Dekker, New York (1993), pp. 111-127.

    Google Scholar 

  120. M. Dugas and R. Göbel, “Every cotorsion-free ring is an endomorphism ring,” Proc. London Math. Soc. (3), 45, No. 2, 319-336 (1982).

    Google Scholar 

  121. M. Dugas and R. Göbel, “Every cotorsion-free algebra is an endomorphism algebra,” Math. Z., 181, 451-470 (1982).

    Google Scholar 

  122. M. Dugas and R. Göbel, “On endomorphism rings of primary Abelian groups,” Math. Ann., 261, No. 3, 359-385 (1982).

    Google Scholar 

  123. M. Dugas and R. G¨obel, “Endomorphism algebras of torsion modules. II,” Lect. Notes Math., 1006, 400-411 (1983).

    Google Scholar 

  124. M. Dugas and R. Göbel, “Almost ?-cyclic Abelian p-groups in L,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 87-105.

    Google Scholar 

  125. M. Dugas and R. Göbel, “Torsion-free Abelian groups with prescribed finitely topologized endomorphism rings,” Proc. Amer. Math. Soc., 90, No. 4, 519-527 (1984).

    Google Scholar 

  126. M. Dugas and R. Göbel, “Endomorphism rings of separable torsion-free Abelian groups,” Houston J. Math., 11, No. 4, 471-483 (1985).

    Google Scholar 

  127. M. Dugas and R. Göbel, “Countable mixed Abelian groups with very nice full-rank subgroups,” Israel J. Math., 51, No. 1-2, 1-12 (1985).

    Google Scholar 

  128. M. Dugas and R. Göbel, “Endomorphism rings of B2-groups of infinite rank,” Israel J. Math., 101, 141-156 (1997).

    Google Scholar 

  129. M. Dugas, R. Göbel, and B. Goldsmith, “Representation of algebras over a complete discrete valuation ring,” Quart. J. Math. Oxford Ser. (2), 35, No. 138, 131-146 (1984).

    Google Scholar 

  130. M. Dugas and J. Hausen, “Torsion-free E-uniserial groups of infinite rank,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 181-189.

    Google Scholar 

  131. M. Dugas, J. Hausen, and J. A. Johnson, “Rings whose additive endomorphisms are ring endomorphisms,” Bull. Austral. Math. Soc., 45, No. 1, 91-103 (1992).

    Google Scholar 

  132. M. Dugas, J. Irwin, and S. Khabbaz, “Countable rings as endomorphism rings,” Quart. J. Math. Oxford Ser. (2), 39, No. 154, 201-211 (1988).

    Google Scholar 

  133. M. Dugas, A. Mader, and C. Vinsonhaler, “Large E-rings exist,” J. Algebra, 108, No. 1, 88-101 (1987).

    Google Scholar 

  134. M. Dugas and B. Olberding, “E-uniserial Abelian groups,” In: Methods in Module Theory (Colorado Springs, CO, 1991), Dekker, New York (1993), pp. 75-85.

    Google Scholar 

  135. M. Dugas and S. Shelah, “E-transitive groups in L,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 191-199.

    Google Scholar 

  136. M. Dugas and B. Thome, “Countable Butler groups and vector spaces with four distinguished subspaces,” J. Algebra, 138, No. 1, 249-272 (1991).

    Google Scholar 

  137. P. C. Eklof, “Set-theoretic methods in homological algebra and Abelian groups,” Press Univ. Montreal, Montreal (1980).

    Google Scholar 

  138. P. C. Eklof and A. H. Mekler, “On endomorphism rings of ú1-separable primary groups,” Lect. Notes Math., 1006, 320-339 (1983).

    Google Scholar 

  139. P. C. Eklof and A. H. Mekler, Almost Free Modules. Set-Theoretic Methods, North-Holland, Amsterdam (1990).

    Google Scholar 

  140. J. N. Fadyn, “The projectivity of Ext(T, A) as a module over E(T),” Pacific J. Math., 85, No. 2, 383-392 (1979).

    Google Scholar 

  141. C. Faith, Algebra: Rings, Modules, and Categories, I, Springer, Berlin (1973).

    Google Scholar 

  142. C. Faith, Algebra. II, Springer, Berlin (1976).

    Google Scholar 

  143. C. Faith and Y. Utumi, “Quasi-injective modules and their endomorphism rings,” Arch. Math., 15, 166-174 (1964).

    Google Scholar 

  144. H. K. Farahat, “Homological dimension and Abelian groups,” Lect. Notes Math., 616, 379-383 (1977).

    Google Scholar 

  145. V. Kh. Farukshin, “Endomorphisms of reduced torsion-free Q p-groups,” In: Numerical Mathematics and Mathematical Physics) [in Russian], Moskov. Gos. Ped. Inst., Moscow (1985), pp. 129-137.

    Google Scholar 

  146. T. Faticoni, “On the lattice of right ideals of the endomorphism ring of an Abelian group,” Bull. Austral. Math. Soc., 38, No. 2, 273-291 (1988).

    Google Scholar 

  147. T. Faticoni, “Gabriel filters on the endomorphism ring of a torsion-free Abelian group,” Comm. Algebra, 18, No. 9, 2841-2883 (1990).

    Google Scholar 

  148. T. Faticoni, “An ew proof of the Baer-Kaplansky theorem,” Comm. Algebra, 19, No. 11, 3119-3123 (1991).

    Google Scholar 

  149. T. Faticoni, “Categories of modules over endomorphism rings,” Mem. Amer. Math. Soc., 103, No. 492 (1993).

  150. T. Faticoni, “Torsion-free Abelian groups torsion over their endomorphism rings,” Bull. Austral. Math. Soc., 50, No. 2, 177-195 (1994).

    Google Scholar 

  151. 151. T. Faticoni, “Examples of torsion-free groups,” Preprint.

  152. T. Faticoni and H. Goeters, “Examples of torsion-free groups flat as modules over their endomorphism rings,” Comm. Algebra, 19, No. 1, 1-27 (1991).

    Google Scholar 

  153. T. Faticoni and P. Schultz, “Direct decompositions of ACD groups with primary regulating index,” In: Abelian Groups and Modules (Colorado Springs, CO, 1995), Dekker, New York (1996), pp. 233-241.

    Google Scholar 

  154. S. Feigelstock, Additive Groups of Rings, Pitman, Boston-London, 1983.

    Google Scholar 

  155. S. T. Files, “Outer automorphisms of endomorphism rings of Warfield groups,” Arch. Math., 65, No. 1, 15-22 (1995).

    Google Scholar 

  156. S. T. Files and W. Wickless, “The Baer-Kaplansky theorem for a class of global mixed groups,” Rocky Mountain J. Math., 26, No. 2, 593-613 (1996).

    Google Scholar 

  157. A. A. Fomin, “Abelian groups with free subgroups of infinite index and their endomorphism rings.,” Mat. Zametki 36, No. 2, 179-187 (1984).

    Google Scholar 

  158. A. A. Fomin, “Torsion-free Abelian groups of infinite rank up to quasi-isomorphism,” In: International Conference on Algebra. Reports on Group Theory [in Russian], Novosibirsk, (1989), P. 128.

  159. A. A. Fomin, “Torsion-free Abelian groups of rank 3,” Mat. Sb., 180, No. 9, 1155-1170 (1989).

    Google Scholar 

  160. A. A. Fomin, “The category of quasi-homomorphisms of Abelian torsion free groups of infinite rank,” In: Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989), Amer. Math. Soc., Providence, RI (1992), pp. 91-111.

    Google Scholar 

  161. B. Franzen and B. Goldsmith, “On endomorphism algebras of mixed modules,” J. London Math. Soc. (2), 31, No. 3, 468-472 (1985).

    Google Scholar 

  162. L. Fuchs, Abelian Groups, Pergamon Press, New York-Oxford-London-Paris (1960).

    Google Scholar 

  163. L. Fuchs, “Recent results and problems on Abelian groups,” In: Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), PUBL: Scott, Foresman and Co., Chicago, Ill. (1963), pp. 9-40.

    Google Scholar 

  164. L. Fuchs, Infinite Abelian Groups. Vol. I, Academic Press, New York-London (1970).

    Google Scholar 

  165. L. Fuchs, Infinite Abelian groups. Vol. II, Academic Press, New York-London (1973).

    Google Scholar 

  166. L. Fuchs, “On torsion Abelian groups quasi-projective over their endomorphism rings,” Proc. Amer. Math. Soc., 42, 13-15 (1974).

    Google Scholar 

  167. L. Fuchs and C. Metelli, “Indecomposable Butler groups of large cardinalities,” Arch. Math., 57, No. 4, 339-344 (1991).

    Google Scholar 

  168. L. Fuchs and K. M. Rangaswamy, “On generalized regular rings,” Math. Z., 107, 71-81 (1968).

    Google Scholar 

  169. L. Fuchs and P. Schultz, “Endomorphism rings of valued vector spaces,” Rend. Sem. Mat. Univ. Padova, 65, 103-110 (1981).

    Google Scholar 

  170. L. Fuchs and P. Schultz, “The Jacobson radical of the endomorphism ring of a valued vector space,” In: Proc. of the First Western Australian Conf. on Algebra, Marcel Dekker, New York (1982), pp. 123-132.

    Google Scholar 

  171. L. Fuchs and G. Viljoen, “Agen eralization of separable torsion-free Abelian groups,” Rend. Sem. Mat. Univ. Padova., 73, 15-21 (1985).

    Google Scholar 

  172. S. A. Gacsalyi, “On limit operations in a certain topology for endomorphism rings of Abelian groups,” Publ. Math. Debrecen, 7, 353-358 (1960).

    Google Scholar 

  173. B. J. Gardner and P. N. Stewart, “Injectives for ring monomorphisms with accessible images. II,” Comm. Algebra, 13, No. 1, 133-145 (1985).

    Google Scholar 

  174. L. Gewirtzman, “Anti-isomorphisms of the endomorphism rings of torsion-free modules,” Math. Z., 98, 391-400 (1967).

    Google Scholar 

  175. S. Glaz and W. Wickless, “Regular and principal projective endomorphism rings of mixed Abelian groups,” Comm. Algebra, 22, No. 4, 1161-1176 (1994).

    Google Scholar 

  176. R. Göbel, “Endomorphism rings of Abelian groups,” Lect. Notes Math., 1006, 340-353 (1983).

    Google Scholar 

  177. R. Göbel, “The existence of rigid systems of maximal size,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 189-202.

    Google Scholar 

  178. R. Göbel, “Modules with distinguished submodules and their endomorphism algebras,” In: Abelian Groups (Curacao, 1991), Dekker, New York (1993), pp. 55-64.

    Google Scholar 

  179. R. Göbel and B. Goldsmith, “Cotorsion-free algebras as endomorphism algebras in L-the discrete and topological cases,” Comment. Math. Univ. Carolin., 34, No. 1, 1-9 (1993).

    Google Scholar 

  180. R. Göbel and W. May, “The construction of mixed modules from torsion modules,” Arch. Math., 62, No. 3, 199-202 (1994).

    Google Scholar 

  181. H. Goeters, “Finitely faithful S-groups,” Comm. Algebra, 17, No. 6, 1291-1302 (1989).

    Google Scholar 

  182. H. Goeters, “Torsion-free Abelian groups with infinite rank endomorphism rings,” Quaest. Math., 14, No. 1, 111-115 (1991).

    Google Scholar 

  183. H. Goeters and J. D. Reid, “On the p-rank of Hom(A,B),” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 171-179.

    Google Scholar 

  184. H. Goeters, C. Vinsonhaler, and W. Wickless, “Agen eralization of quasi-pure injective torsion-free Abelian groups of infinite rank,” Houston J. Math., 22, No. 3, 473-484 (1996).

    Google Scholar 

  185. H. Goeters and W. J. Wickless, “Hyper-ô groups,” Comm. Algebra, 17, No. 6, 1275-1290 (1989).

    Google Scholar 

  186. B. Goldsmith, “Endomorphism rings of torsion-free modules over a complete discrete valuation ring,” J. London Math. Soc. (2), 18, No. 3, 415-417 (1978).

    Google Scholar 

  187. B. Goldsmith, “An essentially semirigid class of modules,” J. London Math. Soc. (2), 29, No. 3, 415-417 (1984).

    Google Scholar 

  188. B. Goldsmith, “Realizing rings as endomorphism rings-the impact of logic,” Irish Math. Soc. Bull., No. 17, 20-28 (1986).

  189. B. Goldsmith, “On endomorphisms and automorphisms of some torsion-free modules,” In: Abelian Group Theory, Gordon and Breach, New York (1987), pp. 417-423.

    Google Scholar 

  190. B. Goldsmith, “On endomorphism rings of nonseparable Abelian p-groups,” J. Algebra, 127, No. 1, 73-79 (1989).

    Google Scholar 

  191. S. Ya. Grinshpon, “Primary Abelian groups with isomorphic endomorphism groups,” Mat. Zametki, 14, No. 5, 733-740 (1973).

    Google Scholar 

  192. S. Ya. Grinshpon, “On the structure of completely characteristic subgroups of Abelian torsion-free groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1982), pp. 56-92.

    Google Scholar 

  193. S. Ya. Grinshpon and V. M. Misyakov, “Completely transitive Abelian groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1986), pp. 12-27.

    Google Scholar 

  194. S. Ya. Grinshpon and V. M. Misyakov, “Some classes of completely transitive torsion-free Abelian groups,” In: Abelian Groups and Modules [in Russian], No. 9, Tomsk. Gos. Univ., Tomsk (1990), pp. 31-36.

    Google Scholar 

  195. S. Ya. Grinshpon and V. M. Misyakov, “Complete transitivity of direct products of Abelian groups,” In: Abelian Groups and Modules [in Russian], No. 10, Tomsk. Gos. Univ., Tomsk (1991), pp. 23-30.

    Google Scholar 

  196. S. Ya. Grinshpon and A. M. Sebeldin, “De.nability of torsion Abelian groups by their groups of endomorphisms,” Mat. Zametki, 57, No. 5, 663-669 (1995).

    Google Scholar 

  197. A. J. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin (1989).

    Google Scholar 

  198. F. Haimo, “Endomorphism radicals which characterize some divisible groups,” Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 10, 25-29 (1967).

    Google Scholar 

  199. F. Haimo, “The Jacobson radical of some endomorphism rings,” In: Studies on Abelian Groups (Symposium, Montpellier, 1967), Springer, Berlin (1968), pp. 143-146.

    Google Scholar 

  200. F. Haimo, “Free subgroups in the endomorphism ring of an Abelian free group,” Portugal. Math., 40 (1981), No. 4, 393-398 (1985).

    Google Scholar 

  201. N. Hart, “Determining groups from endomorphism rings for Abelian groups modulo bounded groups,” Acta Math. Acad. Sci. Hungar., 22 159-162 (1971/1972 ).

    Google Scholar 

  202. G. J. Haupt.eisch, “Torsion free Abelian groups with isomorphic endormorphism rings,” Arch. Math., 24, 269-273 (1973).

    Google Scholar 

  203. J. Hausen, “On automorphism groups and endomorphism rings of Abelian p-groups,” Trans. Amer. Math. Soc., 210, 123-128 (1975).

    Google Scholar 

  204. J. Hausen, “The Jacobson radical of some endomorphism rings,” Lect. Notes Math., 616, 332-336 (1977).

    Google Scholar 

  205. J. Hausen, “The Jacobson radical of endomorphism rings of totally projective groups of finite type,” J. Reine Angew. Math., 292, 19-24 (1977).

    Google Scholar 

  206. J. Hausen, “Quasi-regular ideals of some endomorphism rings,” Ill. J. Math., 21, No. 4, 845-851 (1977).

    Google Scholar 

  207. J. Hausen, “Radicals in endomorphism rings of primary Abelian groups,” In: 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. 63-66.

    Google Scholar 

  208. J. Hausen, “Abelian groups which are uniserial as modules over their endomorphism rings,” Lect. Notes Math., 1006, 204-208 (1983).

    Google Scholar 

  209. J. Hausen, “E-uniserial torsion-free Abelian groups of finite rank,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 181-187.

    Google Scholar 

  210. J. Hausen, “Finite rank torsion-free Abelian groups uniserial over their endomorphism rings,” Proc. Amer. Math. Soc., 93, No. 2, 227-231 (1985).

    Google Scholar 

  211. J. Hausen, “On strongly irreducible torsion-free Abelian groups,” In: Abelian Group Theory (Oberwolfach, 1985), Gordon and Breach, New York (1987), pp. 351-358.

    Google Scholar 

  212. J. Hausen, “E-transitive torsion-free Abelian groups,” J. Algebra, 107, No. 1, 17-27 (1987).

    Google Scholar 

  213. J. Hausen, “Modules with the summand intersection property,” Comm. Algebra, 17, No. 1, 135-148 (1989).

    Google Scholar 

  214. J. Hausen, “Abelian groups whose semi-endomorphisms form a ring,” In: Abelian Groups (Curacao, 1991), Dekker, New York (1993), pp. 175-180.

    Google Scholar 

  215. J. Hausen and J. A. Johnson, “Characterization of the primary Abelian groups, bounded modulo the divisible subgroup, by the radical of their endomorphism rings,” Arch. Math. 29, No. 6, 566-570 (1977).

    Google Scholar 

  216. J. Hausen and J. A. Johnson, “Ideals and radicals of some endomorphism rings,” Pacific J. Math., 74, No. 2, 365-372 (1978).

    Google Scholar 

  217. J. Hausen and J. A. Johnson, “Separability of sufficiently projective p-groups as reflected in their endomorphism rings,” J. Algebra, 69, No. 2, 270-280 (1981).

    Google Scholar 

  218. J. Hausen and J. A. Johnson, “A note on constructing E-rings,” Publ. Math. Debrecen, 38, No. 1-2, 33-38 (1991).

    Google Scholar 

  219. J. Hausen and J. A. Johnson, “Determining Abelian p-groups by the Jacobson radical of their endomorphism rings,” J. Algebra, 174, No. 1, 217-224 (1995).

    Google Scholar 

  220. J. Hausen, C. E. Praeger, and P. Schultz, “Most Abelian p-groups are determined by the Jacobson radical of their endomorphism rings,” Math. Z., 216, No. 3, 431-436 (1994).

    Google Scholar 

  221. I. N. Herstein, Noncommutative Rings, John Wiley and Sons, New York (1968).

    Google Scholar 

  222. P. Hill, “Endomorphism rings generated by units,” Trans. Amer. Math. Soc., 141, 99-105 (1969).

    Google Scholar 

  223. P. Hill, “Errata to: Endomorphism rings generated by units,” Trans. Amer. Math. Soc., 157, 511 (1971).

    Google Scholar 

  224. M. Huber, “On Cartesian powers of a rational group,” Math. Z., 169, No. 3, 253-259 (1979).

    Google Scholar 

  225. M. Huber and R. B. Warfield, “Homomorphisms between Cartesian powers of an Abelian group,” Lect. Notes Math., 874, 202-227 (1981).

    Google Scholar 

  226. A. V. Ivanov, “Abelian groups with self-injective rings of endomorphisms and with rings of endomorphisms with the annihilator condition,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1982), pp. 93-109.

    Google Scholar 

  227. G. Ivanov, “Generalizing the Baer-Kaplansky theorem” (in press)

  228. N. Jacobson, Structure of Rings, Amer. Math. Soc., Providence, Rhode Island (1968).

    Google Scholar 

  229. C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach (1989).

  230. B. Jonsson, “On direct decomposition of torsion free Abelian groups,” Math. Scand., 7, 316-371 (1959).

    Google Scholar 

  231. M. I. Kabenjuk, “Some topologies on Abelian group related to the ring of its endomorphisms,” In: Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), North-Holland, Amsterdam-New York (1980), pp. 705-712.

    Google Scholar 

  232. I. Kaplansky, “Some results on Abelian groups,” Proc. Natl. Acad. Sci. USA, 38, 538-540 (1952).

    Google Scholar 

  233. I. Kaplansky, Infinite Abelian Groups, The University of Michigan Press, Ann Arbor, Mich., (1954, 1969).

    Google Scholar 

  234. I. Kaplansky, “Projective modules,” Ann. Math., 68, 372-377 (1958).

    Google Scholar 

  235. I. Kaplansky and G. Mackey, “A generalization of Ulm's theorem,” Summa Brasil. Math., 2, 195-202 (1951).

    Google Scholar 

  236. F. Kasch, Moduln und Ringe, B. G. Teubner, Stuttgart (1977).

    Google Scholar 

  237. E. Katz and S. A. Morris, “On endomorphisms of Abelian topological groups,” Proc. Amer. Math. Soc., 85, No. 2, 181-183 (1982).

    Google Scholar 

  238. P. Keef, “Slenderness, completions, and duality for primary Abelian groups,” J. Algebra 187, No. 1, 169-182 (1997).

    Google Scholar 

  239. A. Kertesz, Vorlesungen uber artinsche Ringe, Akademiai Kiado, Budapest (1968).

    Google Scholar 

  240. A. Kertesz and T. Szele, “On Abelian groups every multiple of which is a direct summand,” Acta Sci. Math. Szeged, 14, 157-166 (1952).

    Google Scholar 

  241. S. A. Khabbaz and E. H. Toubassi, “The module structure of Ext (F, T) over the endomorphism ring of T,” Pacific J. Math., 54, 169-176 (1974).

    Google Scholar 

  242. S. A. Khabbaz and E. H. Toubassi, “Ext(A, T) as a module over End(T),” Proc. Amer. Math. Soc., 48, 269-275 (1975).

    Google Scholar 

  243. S. F. Kozhukhov, “Regularly complete Abelian groups,” Izv. Vuzov. Mat., No. 12, 14-19 (1980).

  244. M. Krol, “The automorphism groups and endomorphism rings of torsion-free Abelian groups of rank two,” Diss. Math. Rozprawy Mat., 55, 1-76 (1967).

    Google Scholar 

  245. P. A. Krylov, “Radicals of endomorphism rings of torsion-free Abelian groups,” Mat. Sb., 95, No. 2, 214-228 (1974).

    Google Scholar 

  246. P. A. Krylov, “Sums of automorphisms of Abelian groups, and the Jacobson radical of the endomorphism ring,” Izv. Vuzov. Mat., No. 4(167), 56-66 (1976).

    Google Scholar 

  247. P. A. Krylov, “Abelian torsion-free groups with cyclic p-basis subgroups,” Mat. Zametki, 20, No. 6, 805-813 (1976).

    Google Scholar 

  248. P. A. Krylov, “Abelian torsion-free groups and their rings of endomorphisms,” Izv. Vuzov. Mat., No. 11, 26-33 (1979).

  249. P. A. Krylov, “Strongly homogeneous torsion-free Abelian groups,” Sib. Mat. Zh., 24, No. 2, 77-84 (1983).

    Google Scholar 

  250. P. A. Krylov, “On the equivalence of module categories and applications to the theory of Abelian groups,” In: 17th All-Union Algebraic Conference. Part 2 [in Russian], Minsk (1983), pp. 117-118.

  251. P. A. Krylov, “Torsion-free Abelian groups, I,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1984), pp. 40-64.

    Google Scholar 

  252. P. A. Krylov, “Torsion-free Abelian groups, II,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1985), pp. 56-79.

    Google Scholar 

  253. P. A. Krylov, “On the equivalence of categories of Abelian groups and categories of modules over endomorphism rings,” In: 18th All-Union Algebraic Conference. Part 1 [in Russian], Kishinev (1985), P. 288.

  254. P. A. Krylov, “Irreducible torsion-free Abelian groups and their endomorphism rings,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1986), pp. 73-100.

    Google Scholar 

  255. P. A. Krylov, “A class of Abelian groups with hereditary endomorphism rings,” Sib. Mat. Zh., 28, No. 6, 60-65 (1987).

    Google Scholar 

  256. P. A. Krylov, “On endogenerating Abelian groups,” Izv. Vuzov. Mat., No. 12, 52-55 (1987).

  257. P. A. Krylov, “Torsion-free Abelian groups with hereditary endomorphism rings,” Algebra Logika, 27, No. 3, 295-304 (1988).

    Google Scholar 

  258. P. A. Krylov, “Some examples of quasi-pure injective and transitive torsion-free Abelian groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1988), pp. 81-99.

    Google Scholar 

  259. P. A. Krylov, “On one class of quasi-pure injective Abelian groups,” Mat. Zametki, 45, No. 4, 53-58 (1989).

    Google Scholar 

  260. P. A. Krylov, “Torsion-free Abelian groups as flat modules over their endomorphism rings,” In: Abelian Groups and Modules, No. 8 [in Russian], Tomsk. Gos. Univ., Tomsk (1989), pp. 90-96.

    Google Scholar 

  261. P. A. Krylov, “On modules with hereditary endomorphism rings,” Usp. Mat. Nauk, 45, No. 4, 159-160 (1990).

    Google Scholar 

  262. P. A. Krylov, “Locally A-projective torsion-free Abelian groups,” In: Abelian Groups and Modules, No. 9 [in Russian], Tomsk. Gos. Univ., Tomsk (1990), pp. 36-51.

    Google Scholar 

  263. P. A. Krylov, “Completely transitive torsion-free Abelian groups,” Algebra Logika, 29, No. 5, 549-560 (1990).

    Google Scholar 

  264. P. A. Krylov, “Direct sums of strongly homogeneous Abelian groups,” Izv. Vuzov, Mat., No. 2, 65-68 (1991).

  265. P. A. Krylov, “On two problems related to extension groups of Abelian groups,” Mat. Sb., 185, No. 1, 73-94 (1994).

    Google Scholar 

  266. P. A. Krylov, “When is the extension group Ext(A,B) torsion-free?” Izv. Vuzov, Mat., No. 10, 33-41 (1994).

    Google Scholar 

  267. P. A. Krylov, “Jacobson radical of the endomorphism ring of a torsion-free Abelian group,” In: Abelian Groups and Modules, No. 11-12 [in Russian], Tomsk. Gos. Univ., Tomsk (1994), pp. 99-120.

    Google Scholar 

  268. P. A. Krylov, “Torsion-free Abelian groups as modules over their endomorphism rings,” In: Abelian Groups and Modules, No. 13-14 [in Russian], Tomsk. Gos. Univ., Tomsk (1996), pp. 77-104.

    Google Scholar 

  269. P. A. Krylov and E. D. Klassen, “The center of an endomorphism ring of a split mixed Abelian group,” Sib. Mat. Zh., 40, No. 5, 1074-1085 (1999).

    Google Scholar 

  270. P. A. Krylov and E. G. Pakhomova, “The study of the group Hom(A,B) as an injective E(B)-module,” In: Abelian Groups and Modules, No. 13-14 [in Russian], Tomsk. Gos. Univ., Tomsk (1996), pp. 132-169.

    Google Scholar 

  271. P. A. Krylov and E. G. Pakhomova, “Abelian groups as injective modules over their endomorphism rings,” Preprint.

  272. P. A. Krylov and E. I. Podberezina, “The structure of mixed Abelian groups with Noetherian endomorphism rings,” In: Abelian Groups and Modules, No. 11-12 [in Russian], Tomsk. Gos. Univ., Tomsk (1994), pp. 121-128.

    Google Scholar 

  273. P. A. Krylov and E. I. Podberezina, “The group Hom(A,B) as an Artinian E(B)-module,” In: Abelian Groups and Modules, No. 13-14 [in Russian], Tomsk. Gos. Univ., Tomsk (1996), pp. 170-184.

    Google Scholar 

  274. A. G. Kurosh, The Theory of Groups, Chelsea, New York (1960).

    Google Scholar 

  275. E. L. Lady, “Nearly isomorphic torsion free Abelian groups,” J. Algebra, 35, 235-238 (1975).

    Google Scholar 

  276. J. Lambeck, Lectures on Rings and Modules, Blaisdell, Waltham (1968).

    Google Scholar 

  277. D. A. Lawver, “On the commutativity and generalized regularity of ?(G),” Acta Math. Acad. Sci. Hungar, 24, 107-112 (1973).

    Google Scholar 

  278. D. A. Lawver, “Abelian groups in which endomorphic images are fully invariant,” J. Algebra, 29, 232-245 (1974).

    Google Scholar 

  279. L. C. A. van Leeuwen, “On the endomorphism ring of direct sums of groups,” Acta Sci. Math. Szeged, 28 21-29 (1967).

    Google Scholar 

  280. L. C. A. van Leeuwen, “Remarks on endomorphism rings of torsion-free Abelian groups,” Acta Sci. Math. Szeged, 32, 345-350 (1971).

    Google Scholar 

  281. H. Lenzing, “Halberbliche Endomorphismenringe,” Math.-Z., 118, 219-240 (1970).

    Google Scholar 

  282. L. Levy, “Torsion-free and divisible modules over non-integral domains,” Canad. J. Math., 15, 132-151 (1963).

    Google Scholar 

  283. W. Liebert, “Charakterisierung der Endomorphismenringe beschrankter abelscher Gruppen,” Math. Ann., 174, 217-232 (1967).

    Google Scholar 

  284. W. Liebert, “Charakterisierung der Endomorphismenringe endlicher abelscher Gruppen,” Arch. Math. 18, 128-135 (1967).

    Google Scholar 

  285. W. Liebert, “Die minimalen Ideale der Endomorphismenringe Abelscher p-Gruppen,” Math. Z., 97, 85-104, (1967).

    Google Scholar 

  286. W. Liebert, “Endomorphism rings of Abelian p-groups,” In: Studies on Abelian Groups (Symposium, Montpellier, 1967), Springer, Berlin (1968), pp. 239-258.

    Google Scholar 

  287. W. Liebert, “Endomorphism rings of reduced complete torsion-free modules over complete discrete valuation rings,” Proc. Amer. Math. Soc., 36, 375-378 (1972).

    Google Scholar 

  288. W. Liebert, “Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings,” Trans. Amer. Math. Soc., 169, 347-363 (1972).

    Google Scholar 

  289. W. Liebert, “The Jacobson radical of some endomorphism rings,” J. Reine Angew. Math., 262/263, 166-170 (1973).

    Google Scholar 

  290. W. Liebert, “One-sided ideals in the endomorphism rings of reduced complete torsion-free modules and divisible torsion modules over complete discrete valuation rings,” In: Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani, INDAM, Rome, 1972), Academic Press, London (1974), pp. 273-298.

    Google Scholar 

  291. W. Liebert, “Endomorphism rings of free modules over principal ideal domains,” Duke Math. J., 41, 323-328 (1974).

    Google Scholar 

  292. W. Liebert, “Ulm valuations and co-valuations on torsion-complete p-groups,” Lect. Notes Math., 616, 337-353 (1977).

    Google Scholar 

  293. W. Liebert, “Endomorphism rings of Abelian p-groups,” Lect. Notes Math., 1006, 384-399 (1983).

    Google Scholar 

  294. O. Lubimcev, A. M. Sebeldin, and C. Vinsonhaler, “Separable torsion-free Abelian E.-groups,” J. Pure Appl. Algebra, 133, No. 1-2, 203-208 (1998).

    Google Scholar 

  295. A. Mader, “On the automorphism group and the endomorphism ring of Abelian groups,” Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 8, 3-12 (1965).

    Google Scholar 

  296. A. Mader, “On the normal structure of the automorphism group and the ideal structure of the endomorphism ring of Abelian p-groups,” Publ. Math. Debrecen, 13, 123-137 (1966).

    Google Scholar 

  297. A. Mader, “Groups and modules that are slender as modules over their endomorphism rings,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 315-327.

    Google Scholar 

  298. A. Mader and P. Schultz, “Endomorphism rings and automorphism groups of almost completeley decomposable groups,” Research Report, Univ. of Western Australia, 1-21 (1996).

  299. A. Mader and C. Vinsonhaler, “Torsion-free E-modules,” J. Algebra, 115, No. 2, 401-411 (1988).

    Google Scholar 

  300. A. Mader and C. Vinsonhaler, “The idempotent lifting theorem for almost completely decomposable Abelian groups, Colloq. Math., 72, No. 2, 305-317 (1997).

    Google Scholar 

  301. V. T. Markov, A. V. Mikhalev, L. A. Skornyakov, and A. A. Tuganbaev, “Endomorphism rings of modules and lattices of submodules,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 21 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1983), pp. 183-254.

    Google Scholar 

  302. W. May, “Endomorphism rings of Abelian groups with ample divisible subgroups,” Bull. London Math. Soc., 10, No. 3, 270-272 (1978).

    Google Scholar 

  303. W. May, “Endomorphism rings of mixed Abelian groups,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 61-74.

    Google Scholar 

  304. W. May, “Isomorphism of endomorphism algebras over complete discrete valuation rings,” Math. Z., 204, No. 4, 485-499 (1990).

    Google Scholar 

  305. W. May, “Endomorphism algebras of not necessarily cotorsion-free modules,” In: Abelian Groups and Noncommutative Rings, Amer. Math. Soc., Providence, RI (1992), pp. 257-264.

    Google Scholar 

  306. W. May, “Endomorphisms over incomplete discrete valuation domains,” In: Abelian Group Theory and Related Topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI (1994), pp. 277-285.

    Google Scholar 

  307. W. May and E. Toubassi, “Endomorphisms of Abelian groups and the theorem of Baer and Kaplansky,” J. Algebra, 43, No. 1, 1-13 (1976).

    Google Scholar 

  308. W. May and E. Toubassi, “Ar esult on problem 87 of L. Fuchs,” Lect. Notes Math., 616, 354-367 (1977).

    Google Scholar 

  309. W. May and E. Toubassi, “Isomorphisms of endomorphism rings of rank one mixed groups,” J. Algebra, 71, No. 2, 508-514 (1981).

    Google Scholar 

  310. W. May and E. Toubassi, “Endomorphisms of rank one mixed modules over discrete valuation rings,” Pacific J. Math., 108, No. 1, 155-163 (1983).

    Google Scholar 

  311. C. Metelli and L. Salce, “The endomorphism ring of an Abelian torsionfree homogeneous separable group,” Arch. Math., 26, No. 5, 480-485 (1975).

    Google Scholar 

  312. A. V. Mikhalev, “On isomorphisms of rings of continuous endomorphisms,” Sib. Mat. Zh., 4, No. 1, 177-186 (1963).

    Google Scholar 

  313. A. V. Mikhalev, “Isomorphisms of semigroups of endomorphisms of modules,” Algebra Logika, 5, No. 5, 59-67 (1966).

    Google Scholar 

  314. A. V. Mikhalev, “Isomorphisms of semigroups of endomorphisms of modules,” Algebra Logika, 6, No. 2, 35-47 (1967).

    Google Scholar 

  315. A. V. Mikhalev, “Endomorphism rings and submodule lattices,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 12 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1974), pp. 183-254.

    Google Scholar 

  316. A. V. Mikhalev, “Isomorphisms of endomorphism rings of modules that are close to free,” Vestn. MGU I Mat. Mekh., No. 2, 20-27 (1989).

  317. A. V. Mikhalev, “Isomorphisms and anti-isomorphisms of endomorphism rings of modules,” In: First International Tainan-Moscow Algebra Workshop (Tainan, 1994), de Gruyter, Berlin (1996), pp. 69-122.

    Google Scholar 

  318. A. V. Mikhalev and A. P. Mishina, “Infinite Abelian groups: Methods and results,” Fundam. Prikl. Mat., 1, No. 2, 319-375 (1995).

    Google Scholar 

  319. A. P. Mishina, “On automorphisms and endomorphisms of Abelian groups,” Vestn. MGU I Mat. Meh., No. 4, 39-43 (1962).

  320. A. P. Mishina, “Abelian groups,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1967), pp. 9-44.

    Google Scholar 

  321. A. P. Mishina, “Abelian groups,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 10 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1972), pp. 5-45.

    Google Scholar 

  322. A. P. Mishina, “The automorphisms and endomorphisms of Abelian groups,” Vestn. MGU I Mat. Meh., 27, No. 1, 62-66 (1972).

    Google Scholar 

  323. A. P. Mishina, “Abelian groups,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 17 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1979), pp. 3-63.

    Google Scholar 

  324. A. P. Mishina, “Abelian groups,” In: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 23 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1985), pp. 51-118.

    Google Scholar 

  325. A. P. Mishina, “Abelian groups,” J. Math. Sci., 76, No. 6, 2721-2792 (1995).

    Google Scholar 

  326. V. M. Misyakov, “Complete transitivity of reduced Abelian groups,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1993), pp. 134-156.

    Google Scholar 

  327. G. S. Monk, “One-sided ideals in the endomorphism ring of an Abelian p-group,” Acta Math. Acad. Sci. Hungar. 19, 171-185 (1968).

    Google Scholar 

  328. G. S. Monk, “On the endomorphism ring of an Abelian p-group, and of a large subgroup,” Pacific J. Math. 41, 183-193 (1972).

    Google Scholar 

  329. K. Motose, “On Sexauer-Warnock theorem,” J. Fac. Sci. Shinshu Univ., 5, 33-34 (1970).

    Google Scholar 

  330. C. E. Murley, “The classification of certain classes of torsion free Abelian groups,” Pacific J. Math., 40, 647-655 (1972).

    Google Scholar 

  331. C. E. Murley, “Direct products and sums of torsion-free Abelian groups,” Proc. Amer. Math. Soc., 38, 235-241 (1973).

    Google Scholar 

  332. O. Mutzbauer, “Endomorphism rings of Reid groups,” In: Abelian Groups and Modules (Padova, 1994), Kluwer Academic, Dordrecht (1995), pp. 373-383.

    Google Scholar 

  333. O. Mutzbauer, “Endomorphism rings of torsion-free Abelian groups of finite rank,” In: Advances in Algebra and Model Theory (Essen, 1994; Dresden, 1995), Gordon and Breach, Amsterdam (1997), pp. 319-343.

    Google Scholar 

  334. Ngo Dac Tan, “Uber abelsche Gruppen, deren voller Endomorphismenring ein EEkMI-ring (k = 1, 2) ist,” Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 22/23, 75-85 (1979/80).

    Google Scholar 

  335. G. P. Niedzwecki and J. D. Reid, “Abelian groups projective over their endomorphism rings,” J. Algebra, 159, No. 1, 139-143 (1993).

    Google Scholar 

  336. R. J. Nunke, “A note on endomorphism rings of Abelian p-groups,” In: Studies on Abelian Groups (Symposium, Montpellier, 1967), Springer, Berlin (1968), pp. 305-308.

    Google Scholar 

  337. A. Orsatti, “Alcuni gruppi abeliani il cui anello degli endomorfismi e locale,” Rend. Sem. Mat. Univ. Padova, 35, 107-115 (1965).

    Google Scholar 

  338. A. Orsatti, “Su di un problema de T. Szele e J. Szendrei,” Rend. Sem. Mat. Univ. Padova, 35, 171-175 (1965).

    Google Scholar 

  339. A. Orsatti, “A class of rings which are the endomorphism rings of some torsion-free Abelian groups,” Ann. Scuola Norm. Sup. Pisa (3), 23, 143-153 (1969).

    Google Scholar 

  340. B. L. Osofsky, “Noninjective cyclic modules,” Proc. Amer. Math. Soc., 19, 1383-1384 (1968).

    Google Scholar 

  341. E. G. Pakhomova, “The group Hom(A,B) as injective module over the endomorphism ring,” In: Symposium on Abelian groups [in Russian], Biisk (1994), P. 21.

  342. A. T. Paras, “Abelian groups as Noetherian modules over their endomorphism rings,” In: Abelian Group Theory and Related Topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI (1994), pp. 325-332.

    Google Scholar 

  343. A. T. Paras, “E-definitely generated groups,” Comm. Algebra 23, No. 13, 4749-4756 (1995).

    Google Scholar 

  344. J. T. Parr, “Endomorphism rings of rank two torsion-free Abelian groups,” Proc. London Math. Soc. (3), 22, 611-632 (1971).

    Google Scholar 

  345. E. M. Patterson, “On the radicals of certain rings of infinite matrices,” Proc. Roy. Soc. Edinburgh Sect. A, 65, 263-271 (1960).

    Google Scholar 

  346. E. M. Patterson, “On the radicals of rings of row-finite matrices,” Proc. Roy. Soc. Edinburgh Sect. A, 66, 42-46 (1961/1962).

    Google Scholar 

  347. R. S. Pierce, “Homomorphisms of primary Abelian groups,” In: Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill. (1963), pp. 215-310.

    Google Scholar 

  348. R. S. Pierce, “Endomorphism rings of primary Abelian groups,” In: Proc. Colloq. Abelian Groups (Tihany, 1963), Akademiai Kiado, Budapest (1964), pp. 125-137.

    Google Scholar 

  349. R. S. Pierce, “E-modules,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 221-240.

    Google Scholar 

  350. R. S. Pierce and C. Vinsonhaler, “Realizing algebraic number fields,” Lect. Notes Math., 1006, 49-96 (1983).

    Google Scholar 

  351. R. S. Pierce and C. Vinsonhaler, “Carriers of torsion-free groups,” Rend. Sem. Mat. Univ. Padova., 84 (1990), 263-281 (1991).

    Google Scholar 

  352. R. S. Pierce and C. Vinsonhaler, “Classifying E-rings,” Comm. Algebra, 19, No. 2, 615-653 (1991).

    Google Scholar 

  353. B. I. Plotkin, Groups of Automorphisms of Algebraic Systems, Wolters-Noordho., Groningen, (1972).

    Google Scholar 

  354. E. I. Podberezina, “The structure of separable torsion-free Abelian groups with Noetherian endomorphism rings,” In: Abelian Groups and Modules, No. 9 [in Russian], Tomsk. Gos. Univ., Tomsk (1990), pp. 77-83.

    Google Scholar 

  355. E. I. Podberezina, “On the Artinianity of the E(A)-module Hom(A,B),” In: Abelian Groups and Modules, No. 13-14 [in Russian], Tomsk. Gos. Univ., Tomsk (1996), pp. 190-199.

    Google Scholar 

  356. G. D. Poole and J. D. Reid, “Abelian groups quasi-injective over their endomorphism rings,” Canad. J. Math. 24, 617-621 (1972).

    Google Scholar 

  357. V. Popa, “Units, idempotents and nilpotents of an endomorphism ring. II,” Izv. Akad. Nauk Respub. Moldova Mat., No. 1, 93-105 (1997).

  358. C. E. Praeger and P. Schultz, “The Loewy length of the Jacobson radical of a bounded endomorphism ring,” In: Abelian Groups and Noncommutative Rings, Amer. Math. Soc., Providence, RI (1992), pp. 349-360.

    Google Scholar 

  359. G. E. Puninski and A. A. Tuganbaev, Rings and Modules, Soyuz, Moscow (1998).

    Google Scholar 

  360. P. Puusemp, “On the determination of a torsion Abelian group by its semigroup of endomorphisms in the class of all periodic Abelian groups,” Eesti NSV Tead. Akad. Toimetised Fuus. Mat., 29, No. 3, 246-253 (1980).

    Google Scholar 

  361. P. Puusemp, “On a theorem of May,” Eesti NSV Tead. Akad. Toimetised Fuus. Mat., 38, No. 2, 139-145 (1989).

    Google Scholar 

  362. P. Puusemp, “On the torsion subgroups and endomorphism semigroups of Abelian groups,” Algebras Groups Geom., 14, No. 4, 407-422 (1997).

    Google Scholar 

  363. K. M. Rangaswamy, “Abelian groups with endomorphic images of special types,” J. Algebra, 6, 271-280 (1967).

    Google Scholar 

  364. K. M. Rangaswamy, “Representing Baer rings as endomorphism rings,” Math. Ann. 190, 167-176 (1970/1971).

    Google Scholar 

  365. K. M. Rangaswamy, “Abelian groups with self-injective endomorphism rings,” Lect. Notes Math., 372, 595-604 1974.

    Google Scholar 

  366. K. M. Rangaswamy, “Separable Abelian groups as modules over their endomorphism rings,” Proc. Amer. Math. Soc., 91, No. 2, 195-198 (1984).

    Google Scholar 

  367. J. D. Reid, “On the ring of quasi-endomorphisms of a torsion-free group,” In: Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill. (1963), pp. 51-68.

    Google Scholar 

  368. J. D. Reid, “On subcommutative rings,” Acta Math. Acad. Sci. Hungar., 16, 23-26 (1965).

    Google Scholar 

  369. J. D. Reid, “Abelian groups finitely generated over their endomorphism rings,” Lect. Notes Math., 874, 41-52 (1981).

    Google Scholar 

  370. J. D. Reid, “Abelian groups cyclic over their endomorphism rings,” Lect. Notes Math., 1006, 190-203 (1983).

    Google Scholar 

  371. J. D. Reid, “Warfield duality and irreducible groups,” In: Abelian Groups and Noncommutative Rings, Amer. Math. Soc., Providence, RI (1992), pp. 361-370.

    Google Scholar 

  372. J. D. Reid and G. Niedzwicki, “Torsion-free Abelian groups cyclic projective over their endomorphism rings,” Preprint.

  373. F. Richman, “Aclas s of rank-2 torsion free groups,” In: Studies on Abelian Groups (Symposium, Montpellier, 1967), Springer, Berlin (1968), pp. 327-333.

    Google Scholar 

  374. F. Richman, “Detachable p-groups and quasi-injectivity,” Acta Math. Acad. Sci. Hungar., 27, No. 1-2, 71-73 (1976).

    Google Scholar 

  375. F. Richman, “An extension of the theory of completely decomposable torsion-free Abelian groups,” Trans. Amer. Math. Soc., 279, No. 1, 175-185 (1983).

    Google Scholar 

  376. F. Richman and E. A. Walker, “Primary Abelian groups as modules over their endomorphism rings,” Math. Z., 89, 77-81 (1965).

    Google Scholar 

  377. F. Richman and E. A. Walker, “Modules over PIDs that are injective over their endomorphism rings,” In: Ring Theory (Proc. Conf., Park City, Utah, 1971), Academic Press, New York (1972), pp. 363-372.

    Google Scholar 

  378. F. Richman and E. A. Walker, “Homological dimension of Abelian groups over their endomorphism rings,” Proc. Amer. Math. Soc., 54, 65-68 (1976).

    Google Scholar 

  379. F. Richman and E. Walker, “Cyclic Ext,” Rocky Mountain J. Math., 11, No. 4, 611-615 (1981).

    Google Scholar 

  380. S. K. Rososhek and M. A. Turmanov, “Periodic Abelian groups that are purely simple as modules over their endomorphism rings,” In: Abelian Groups and Modules, No. 7 [in Russian], Tomsk. Gos. Univ., Tomsk (1988), pp. 106-109.

    Google Scholar 

  381. S. V. Rychkov, “Endomorphism rings of Abelian groups,” Algebra Logika, 27, No. 3, 327-342 (1988).

    Google Scholar 

  382. S. V. Rychkov, “Realization of rings by endomorphism rings,” Algebra Logika, 29, No. 1, 47-66 (1990).

    Google Scholar 

  383. L. Salce and F. Menegazzo, “Abelian groups whose endomorphism ring is linearly compact,” Rend. Sem. Mat. Univ. Padova., 53, 315-325 (1975).

    Google Scholar 

  384. A. D. Sands, “The radical of a certain infinite matrix ring,” Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 12, 143-145 (1969).

    Google Scholar 

  385. A. D. Sands, “On the radical of the endomorphism ring of a primary Abelian group,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 305-314.

    Google Scholar 

  386. P. Schultz, “The endomorphism ring of the additive group of a ring,” J. Austral. Math. Soc., 15, 60-69 (1973).

    Google Scholar 

  387. P. Schultz, “On a paper of Szele and Szendrei on groups with commutative endomorphism rings,” Acta Math. Acad. Sci. Hungar, 24, 59-63 (1973).

    Google Scholar 

  388. P. Schultz, “The endomorphism ring of a valuated group,” In: Abelian Group Theory (Perth, 1987), Amer. Math. Soc., Providence, RI (1989), pp. 75-84.

    Google Scholar 

  389. P. Schultz, “A counter example in Jacobson radicals,” Proc. Amer. Math. Soc., 122, No. 3, 961-964 (1994).

    Google Scholar 

  390. P. Schultz, “When is an Abelian p-group determined by the Jacobson radical of its endomorphism ring?” In: Abelian Group Theory and Related Topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI (1994), pp. 385-396.

    Google Scholar 

  391. A. M. Sebeldin, “Isomorphism conditions of completely decomposable torsion-free abelian groups with isomorphic endomorphism rings,” Mat. Zametki, 11, No. 4, 403-408 (1972).

    Google Scholar 

  392. A. M. Sebeldin, “Complete direct sums of torsion-free Abelian groups of rank 1 with isomorphic groups or rings of endomorphisms, I,” Mat. Zametki, 14, 867-878 (1973).

    Google Scholar 

  393. A. M. Sebeldin, “Complete direct sums of torsion-free Abelian groups of rank 1 with isomorphic groups or rings of endomorphisms, II,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1979), pp. 151-156.

    Google Scholar 

  394. A. M. Sebeldin, “Torsion-free Abelian groups with isomorphic endomorphism rings,” In: Abelian Groups and Modules [in Russian], Tomsk. Gos. Univ., Tomsk (1979), pp. 157-162.

    Google Scholar 

  395. A. M. Sebeldin, “Determination of Abelian groups by their endomorphism rings,” In: Abelian Groups and Modules, No. 8 [in Russian], Tomsk. Gos. Univ., Tomsk (1989), pp. 113-123.

    Google Scholar 

  396. A. M. Sebeldin, “Determination of Abelian groups by their semigroups of endomorphisms,” In: Abelian Groups and Modules, No. 10 [in Russian], Tomsk. Gos. Univ., Tomsk (1991), pp. 125-133.

    Google Scholar 

  397. A. M. Sebeldin, “Determination of vector groups by endomorphism semigroups,” Algebra Logika 33, No. 4, 422-428 (1994).

    Google Scholar 

  398. A. M. Sebeldin, “Abelian groups with isomorphic endomorphism semigroups,” Usp. Mat. Nauk, 49, No. 6, 211-212 (1994).

    Google Scholar 

  399. A. M. Sebeldin, “Abelian groups of certain classes with isomorphic endomorphism rings,” Usp. Mat. Nauk, 50, No. 1(301), 207-208 (1995).

    Google Scholar 

  400. A. M. Sebeldin, “Determination of separable torsion-free Abelian groups by endomorphism semigroups,” Algebra Logika 34, No. 5, 523-530 (1995).

    Google Scholar 

  401. A. M. Sebeldin, “Determination of Abelian groups by their endomorphism rings,” In: Algebra (Krasnoyarsk, 1993), de Gruyter, Berlin (1996), pp. 217-223.

    Google Scholar 

  402. A. M. Sebeldin, “Summability of the endomorphism ring of vector groups,” Algebra Logika, 37, No. 1, 88-100 (1998).

    Google Scholar 

  403. N. E. Sexauer and J. E. Warnock, “The radical of the row-finite matrices over an arbitrary ring,” Trans. Amer. Math. Soc., 139, 287-295 (1969).

    Google Scholar 

  404. S. Shelah, “Infinite Abelian groups, the Whitehead problem and some constructions,” Israel J. Math., 18, 243-256 (1974).

    Google Scholar 

  405. S. Shelah, “A combinatorial principle and endomorphism rings. I. On p-groups,” Israel J. Math., 49, No. 1-3, 239-257 (1984).

    Google Scholar 

  406. S. Shelah, “A combinatorial theorem and endomorphism rings of Abelian groups. II,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 37-86.

    Google Scholar 

  407. L. W. Small, “Semihereditary rings,” Bull. Amer. Math. Soc., 73, 656-658 (1967).

    Google Scholar 

  408. R. W. Stringall, “Endomorphism rings of Abelian groups generated by automorphism groups,” Acta Math. Acad. Sci. Hungar, 18, 401-404 (1967).

    Google Scholar 

  409. R. W. Stringall, “Endomorphism rings of primary Abelian groups,” Pacific J. Math., 20, 535-557 (1967).

    Google Scholar 

  410. C. Szasz, “Die Definition des Ringes der Endomorphismen einer Abelschen Gruppe in einem elementaren Topos,” Bul. Univ. Brasov Ser. C, 26 (1984), 47-50 (1985).

    Google Scholar 

  411. T. Szele, “Uber die Abelschen Gruppen mit nullteilerfreiem Endomorphismenring,” Publ. Math. Debrecen, 1, 89-91 (1949).

    Google Scholar 

  412. T. Szele, “Gruppentheoretishe Beziehungen der Primk¨orper,” Mat. Aineiden Aikakauskirja, 13, 80-85 (1949).

    Google Scholar 

  413. T. Szele, “On a topology in endomorphism rings of Abelian groups,” Publ. Math. Debrecen, 5, 1-4 (1957).

    Google Scholar 

  414. T. Szele and J. Szendrei, “On Abelian groups with commutative endormorphism ring,” Acta Math. Acad. Sci. Hungar., 2, 309-324 (1951).

    Google Scholar 

  415. I. Szelpal, “The Abelian groups with torsion-free endomorphism ring,” Publ. Math. Debrecen, 3 (1953), 106-108 (1954).

    Google Scholar 

  416. E. V. Tarakanov, “Classes of Abelian groups with a certain property of endomorphisms,” In: Abelian Groups and Modules, No. 7 [in Russian], Tomsk. Gos. Univ., Tomsk (1988), pp. 121-123.

    Google Scholar 

  417. B. Thome, “\(\aleph _{\text{1}} \)-separable groups and Kaplansky's test problems,” Forum Math., 2, No. 3, 203-212 (1990).

    Google Scholar 

  418. E. Toubassi and W. May, “Classifying endomorphism rings of rank one mixed groups,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 253-263.

    Google Scholar 

  419. A. A. Tuganbaev, “Distributive rings and modules,” Tr. Mosk. Mat. Obshch., 51, 95-113 (1988).

    Google Scholar 

  420. A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht/ Boston/London (1998).

    Google Scholar 

  421. A. A. Tuganbaev, Distributive Modules and Related Topics, Gordon and Breach, Amsterdam (1999).

    Google Scholar 

  422. M. A. Turmanov, “Separable torsion-free groups as modules over their endomorphism rings,” In: Abelian Groups and Modules, No. 8 [in Russian], Tomsk. Gos. Univ., Tomsk (1989), pp. 128-138.

    Google Scholar 

  423. M. A. Turmanov, “Endopure submodules of torsion-free Abelian groups of rank 2,” In: Abelian Groups and Modules, No. 9 [in Russian], Tomsk. Gos. Univ., Tomsk (1990), pp. 119-124.

    Google Scholar 

  424. M. A. Turmanov, “Endopure submodules of certain Abelian groups,” In: IIIrd International Conference on Algebra, Krasnoyarsk (1993), pp. 336-337.

  425. C. Vinsonhaler, “Torsion free Abelian groups quasi-projective over their endomorphism rings. II,” Pacific J. Math., 74, No. 1, 261-265 (1978).

    Google Scholar 

  426. C. Vinsonhaler, “Corrections to: Torsion free Abelian groups quasi-projective over their endomorphism rings. II,” Pacific J. Math., 79, No. 2, 564-565 (1978).

    Google Scholar 

  427. C. Vinsonhaler, “The divisible and E-injective hulls of a torsion free group,” In: Abelian Groups and Modules (Udine, 1984), Springer, Vienna (1984), pp. 163-179.

    Google Scholar 

  428. C. Vinsonhaler and W. J. Wickless, “Torsion free Abelian groups quasi-projective over their endomorphism rings,” Pacific J. Math., 68, No. 2, 527-535 (1977).

    Google Scholar 

  429. C. Vinsonhaler and W. J. Wickless, “Injective hulls of torsion-free Abelian groups as modules over their endomorphism rings,” J. Algebra, 58, No. 1, 64-69 (1979).

    Google Scholar 

  430. C. Vinsonhaler and W. Wickless, “The injective hull of a separable p-group as an E-module,” J. Algebra, 71, No. 1, 32-39 (1981).

    Google Scholar 

  431. C. Vinsonhaler and W. Wickless, “Locally irreducible rings,” Bull. Austral. Math. Soc., 32, No. 1, 129-145 (1985).

    Google Scholar 

  432. C. Vinsonhaler and W. Wickless, “Dualities for torsion-free Abelian groups of finite rank,” J. Algebra, 128, No. 2, 474-487 (1990).

    Google Scholar 

  433. C. Vinsonhaler and W. Wickless, “Homological dimensions of completely decomposable groups,” In: Abelian Groups (Curacao, 1991), Dekker, New York (1993), pp. 247-258.

    Google Scholar 

  434. C. Vinsonhaler and W. Wickless, “Realizations of finite dimensional algebras over the rationals,” Rocky Mountain J. Math., 24, No. 4, 1553-1565 (1994).

    Google Scholar 

  435. C. L. Walker, “Local-quasi-endomorphism rings of rank one mixed Abelian groups,” Lect. Notes Math., 616 368-378 (1977).

    Google Scholar 

  436. E. A. Walker, “Quotient categories and quasi-isomorphisms of Abelian groups,” In: Proc. Colloq. Abelian Groups (Tihany, 1963), Akademiai Kiado, Budapest (1964), pp. 147-162.

    Google Scholar 

  437. R. Ware, “Endomorphism rings of projective modules,” Trans. Amer. Math. Soc., 155, 233-256 (1971).

    Google Scholar 

  438. R. Ware and J. Zelmanowitz, “The Jacobson radical of the endomorphism ring of a projective module,” Proc. Amer. Math. Soc., 26 15-20 (1970).

    Google Scholar 

  439. R. B. Warfield, “Homomorphisms and duality for torsion-free groups,” Math. Z., 107, 189-200 (1968).

    Google Scholar 

  440. M. C. Webb, “The endomorphism ring of pointed separable torsion-free Abelian groups,” J. Algebra, 55, No. 2, 446-454 (1978).

    Google Scholar 

  441. M. C. Webb, “The endomorphism ring of homogeneously decomposable separable groups,” Arch. Math., 31, No. 3, 235-243 (1978/79).

    Google Scholar 

  442. W. J. Wickless, “T as an ? submodule of G,” Pacific J. Math., 83, No. 2, 555-564 (1979).

    Google Scholar 

  443. W. J. Wickless, “A functor from mixed groups to torsion-free groups,” In: Abelian Group Theory and Related Topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI (1994), pp. 407-417.

    Google Scholar 

  444. W. J. Wickless, “The Baer-Kaplansky theorem to direct sums of self-small mixed groups,” In: Abelian Groups and Modules (Dublin, 1998), Birkäuser, Basel (1999), pp. 101-106.

    Google Scholar 

  445. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, PA (1991).

    Google Scholar 

  446. K. G. Wolfson, “Isomorphisms of the endomorphism rings of torsion-free modules,” Proc. Amer. Math. Soc., 13, 712-714 (1962).

    Google Scholar 

  447. O. Zariski and P. Samuel, Commutative Algevra, Van Nostrand, Princeton (1958).

    Google Scholar 

  448. H. Zassenhaus, “Orders as endomorphism rings of modules of the same rank,” J. London Math. Soc., 42, 180-182 (1967).

    Google Scholar 

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  1. P. A. Krylov
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  2. A. V. Mikhalev
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  3. A. A. Tuganbaev
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Krylov, P.A., Mikhalev, A.V. & Tuganbaev, A.A. Properties of endomorphism rings of Abelian groups, I. Journal of Mathematical Sciences 112, 4598–4735 (2002). https://doi.org/10.1023/A:1020582507609

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  • DOI: https://doi.org/10.1023/A:1020582507609

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