Skip to main content
Log in

Rings Whose Nonzero Modules Have Maximal Submodules

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

REFERENCES

  1. J. S. Alin and E. P. Armendariz, “A class of rings having all singular simple modules injective,” Math. Scand., 23, 233-240 (1968).

    Google Scholar 

  2. N. M. Al-Thani, “Pure Baer injective modules,” Int. J. Math. Math. Sci.No. 3, 529-538 (1997).

  3. E. P. Armendariz, “On semiprime P.I.-algebras over commutative regular rings,” Pacific J. Math.66, No. 1, 23-38 (1976).

    Google Scholar 

  4. E. P. Armendariz, “Modules with Artinian prime factors,” Proc. Amer. Math. Soc., 78, No. 3, 311-314 (1980).

    Google Scholar 

  5. E. P. Armendariz and J. W. Fisher, “Regular P.I.-rings,” Proc. Amer. Math. Soc., 39, 247-251 (1973).

    Google Scholar 

  6. E. P. Armendariz, J. W. Fisher, and S. A. Steinberg, “Central localization of regular rings,” Proc. Amer. Math. Soc., 46, No. 3, 315-321 (1974).

    Google Scholar 

  7. A. G. Athanassiadis, “A note on V-rings,” Bull. Soc. Math. Greece (N.S.), 12, No. 1, 91-95 (1971).

    Google Scholar 

  8. G. Baccella, “Generalized V-rings and von Neumann regular rings,” Rend. Sem. Mat. Univ. Padova, 72, 117-133 (1984).

    Google Scholar 

  9. G. Baccella, “Von Neumann regularity of V-rings with Artinian primitive factor rings,” Proc. Amer. Math. Soc., 103, No. 3, 747-749 (1988).

    Google Scholar 

  10. G. Baccella, “Semi-Artinian V-rings and semi-Artinian von Neumann regular rings,” J. Algebra, 173, No. 3, 587-612 (1995).

    Google Scholar 

  11. H. Bass, “Finistic dimension and a homological generalization of semiprimary rings,” Trans. Amer. Math. Soc., 95, No. 3, 466-488 (1960).

    Google Scholar 

  12. A. K. Boyle, “Hereditary QI-rings,” Trans. Amer. Math. Soc., 192, 115-120 (1974).

    Google Scholar 

  13. A. K. Boyle, “Injectives containing no proper quasi-injective submodules,” Commun. Algebra4, No. 8, 775-785 (1976).

    Google Scholar 

  14. A. K. Boyle and K. R. Goodearl, “Rings over which certain modules are injective,” Pacific J. Math., 58, No. 1, 43-53 (1975).

    Google Scholar 

  15. S. H. Brown, “Rings over which every simple module is rationally complete,” Can. J. Math., 25, 693-701 (1973).

    Google Scholar 

  16. W. D. Burgess and W. Stephenson, “An analogue of the Pierce sheaf for noncommutative rings,” Commun. Algebra6, No. 9, 863-886 (1978).

    Google Scholar 

  17. K. A. Byrd, “Rings whose quasi-injective modules are injective,” Proc. Amer. Math. Soc., 33, 235-240 (1972).

    Google Scholar 

  18. K. A. Byrd, “When are quasi-injectives injective?” Can. Math. Bull., 15, 599-600 (1972).

    Google Scholar 

  19. V. P. Camillo, “On some rings whose modules have maximal submodules,” Proc. Amer. Math. Soc., 50, 97-100 (1975).

    Google Scholar 

  20. V. P. Camillo and K. R. Fuller, “A note on Loewy rings and chain conditions on primitive ideals,” In: Lect. Notes Math., Vol. 700 (1979), pp. 75-86.

    Google Scholar 

  21. V. Camillo and M. F. Yousif, “Semi-V-modules,” Commun. Algebra, 17, No. 1, 165-177 (1989).

    Google Scholar 

  22. J. L. Chen and N. Q. Ding, “On a generalization of V-rings and SF-rings,” Kobe J. Math., 11, No. 1, 101-105 (1994).

    Google Scholar 

  23. D. P. Choudhury and K. Tewari, “Tensor purities, cyclic quasi-projectives, and cocyclic copurity,” Commun. Algebra, 7, No. 15, 1559-1572 (1979).

    Google Scholar 

  24. F. Couchot, “Classes d'anneaux contenant les V-anneaux et les anneaux absolument plats,” In: Lect. Notes Math., Vol. 740 (1979), pp. 170-183.

    Google Scholar 

  25. J. H. Cozzens, “Homological properties of the ring of differential polynomials,” Bull. Amer. Math. Soc., 76, 75-79 (1970).

    Google Scholar 

  26. J. H. Cozzens and C. Faith, Simple Noetherian Rings, Cambridge University Press, Cambridge (1975).

    Google Scholar 

  27. R. F. Damiano, “A right PCI ring is right Noetherian,” Proc. Amer. Math. Soc., 77, No. 1, 11-14 (1979).

    Google Scholar 

  28. N. Q. Ding and J. L. Chen, “Rings whose simple singular modules are YJ-injective,” Math. Jpn., 40, No. 1, 191-195 (1994).

    Google Scholar 

  29. F. Dischinger, “Sur les anneaux fortement Π-réguliers,” C. R. Acad. Sci., 283, No. 8, A571-A573 (1976).

    Google Scholar 

  30. X. Du and J. L. Zhang, “MELT right V-rings are von Neumann regular,” Chinese Sci. Bull., 40, No. 11, 967-968 (1995).

    Google Scholar 

  31. N. V. Dung and P. F. Smith, “On semi-Artinian V-modules,” J. Pure Appl. Algebra, 82, No. 1, 27-37 (1992).

    Google Scholar 

  32. C. Faith, “Lectures on injective modules and quotient rings,” In: Lect. Notes Math., Vol. 49 (1967), pp. 1-140.

    Google Scholar 

  33. C. Faith, “On hereditary rings and Boyle's conjecture,” Arch. Math., 27, No. 2, 113-119 (1976).

    Google Scholar 

  34. C. Faith, “Rings whose modules have maximal submodules,” Publ. Mat., 39, No. 1, 201-214 (1995).

    Google Scholar 

  35. C. Faith, “Locally perfect commutative rings are those whose modules have maximal submodules,” Commun. Algebra, 23, No. 13, 4885-4886 (1995).

    Google Scholar 

  36. C. Faith and P. Menal, “A new duality theorem for semisimple modules and characterization of Villamayor rings,” Proc. Amer. Math. Soc., 123, No. 6, 1635-1637 (1995).

    Google Scholar 

  37. J. W. Fisher, “Von Neumann regular rings versus V-rings,” In: Ring Theory. Proc. Conf. Univ. Oklahoma, 1973, Dekker, New York (1974), pp. 101-119.

    Google Scholar 

  38. J. W. Fisher and R. L. Snider, “On the Von Neumann regularity of rings with regular prime factor rings,” Pacif. J. Math., 54, No. 1, 135-144 (1974).

    Google Scholar 

  39. D. R. Farkas and R. L. Snider, “Group algebras whose simple modules are injective,” Trans. Amer. Math. Soc., 194, 241-248 (1974).

    Google Scholar 

  40. J. L. Garsia Hernandez, and J. L. Gomez Pardo, “V-rings relative to Gabriel topologies,” Commun. Algebra, 13, No. 1, 59-83 (1985).

    Google Scholar 

  41. S. C. Goel, S. K. Jain, and S. Singh, “Rings whose cyclic modules are injective or projective,” Proc. Amer. Math. Soc., 53, No. 1, 16-18 (1975).

    Google Scholar 

  42. J. M. Goursaud, “Sur les V-anneaux reguliers,” In: Seminaire d'Algebre Non Commutative (annee 1975-1976), Exp. No. 5. Publ. Math. Orsay, Nos. 186-7655, U. E. R. Math., Univ. ParisXI, Orsay (1976).

  43. J. M. Goursaud and J. Valette, “Sur l'enveloppe injective des anneaux de groupes reguliers,” Bull. Soc. Math. France, 103, No. 1, 91-102 (1975).

    Google Scholar 

  44. J. S. Golan and Z. Papp, “Cocritically nice rings and Boyle's conjecture,” Commun. Algebra8(18), 1775-1798 (1980).

    Google Scholar 

  45. R. M. Hamsher, “Commutative Noetherian rings over which every module has a maximal submodule,” Proc. Amer. Math. Soc., 17, 1471-1472 (1966).

    Google Scholar 

  46. R. M. Hamsher, “Commutative rings over which every module has a maximal submodule,” Proc. Amer. Math. Soc., 18, 1133-1137 (1967).

    Google Scholar 

  47. F. Hansen, “Certain overrings of right hereditary, right Noetherian rings are V-rings,” Proc. Amer. Math. Soc., 52, 85-90 (1975).

    Google Scholar 

  48. F. Hansen, “On one-sided prime ideals,” Pacific J. Math., 58, No. 1, 79-85 (1975).

    Google Scholar 

  49. Y. Hirano, “Regular modules and V-modules,” In: Proc. 13th Symp. Ring Theory. Okayama Univ. 1980Okayama Univ., Okayama (1981), pp. 24-43.

    Google Scholar 

  50. Y. Hirano, “Regular modules and V-modules,” Hiroshima Math. J., 11, No. 1, 125-142 (1981).

    Google Scholar 

  51. Y. Hirano, “Regular modules and V-modules. II,” Math. J. Okayama Univ., 23, No. 2, 131-135 (1981).

    Google Scholar 

  52. Y. Hirano and H. Tominaga, “Regular rings, V-rings, and their generalizations,” Hiroshima Math. J., 9, No. 1, 137-149 (1979).

    Google Scholar 

  53. V. A. Hiremath, “Coflat modules,” Indian J. Pure Appl. Math., 17, No. 2, 223-230 (1986).

    Google Scholar 

  54. X. H. Hu, “Quasi-regular rings and regular rings,” J. Math. (Wuhan)14, No. 4, 519-522 (1994).

    Google Scholar 

  55. D. V. Huynh, S. K. Jain, and S. R. Lopez-Permouth, “On a class of non-Noetherian V-rings,” Commun. Algebra, 24, No. 9, 2839-2850 (1996).

    Google Scholar 

  56. D. V. Huynh, P. F. Smith, and R. Wisbauer, “A note on GV-modules with Krull dimension,” Glasgow Math. J., 32, No. 3, 389-390 (1990).

    Google Scholar 

  57. A. Idelhadj and E. A. Kaidi, “The dual of the socle-fine notion and applications,” In: Commutative Ring Theory, Dekker, New York (1997), pp. 359-367.

    Google Scholar 

  58. S. K. Jain, S. R. Lopez-Permouth, and L. H. Rowen, “Superfluous covers,” Commun. Algebra, 23, No. 5, 1663-1677 (1995).

    Google Scholar 

  59. N. K. Kim, S. B. Nam, and J. Y. Kim, “On simple singular GP-injective modules,” Commun. Algebra, 27, No. 5, 2087-2096 (1999).

    Google Scholar 

  60. M. H. Klin, “V-rings and their connection with the automorphism groups of binary relations,” Mat. Issled., 7, No. 1, 204-205 (1972).

    Google Scholar 

  61. L. A. Koifman, “Rings over which each module has a maximal submodule,” Mat. Zametki, 7, 359-367 (1970).

    Google Scholar 

  62. K. A. Kosler, “On hereditary rings and Noetherian V-rings,” Pacific J. Math., 103, No. 2, 467-473 (1982).

    Google Scholar 

  63. Z. K. Liu, “Characterization for two kinds of rings by using the quasi-injectivity of some modules,” Northeast. Math. J., 7, No. 2, 141-149 (1991).

    Google Scholar 

  64. Z. K. Liu, “A characterization of F-V-rings by generalized injectivity,” Acta Math. Sinica38, No. 2, 200-206 (1995).

    Google Scholar 

  65. Z. K. Liu and J. Ahsan, “Characterizations of F-V-rings by quasi-continuous modules,” Northeast. Math. J., 11, No. 4, 417-424 (1995).

    Google Scholar 

  66. V. T. Markov, “On B-rings with a polynomial identity,” Tr. Semin. Petrovskogo, 7, 232-238 (1981).

    Google Scholar 

  67. G. O. Michler and O. E. Villamayor, “On rings whose simple modules are injective,” J. Algebra, 25, 185-201 (1973).

    Google Scholar 

  68. S. Mohamed, “On PCI rings,” J. Univ. Kuwait Sci., 2, 21-23 (1975).

    Google Scholar 

  69. S. B. Nam, N. K. Kim, and J. Y. Kim, “On simple GP-injective modules,” Commun. Algebra, 23, No. 14, 5437-5444 (1995).

    Google Scholar 

  70. C. Nastasescu, “ Quelques remarques sur la dimension homologique des anneaux,” J. Algebra, 19, 470-485 (1971).

    Google Scholar 

  71. K. Ohtake, “Commutative rings of which all radicals are left exact,” Commun. Algebra8, No. 16, 1505-1512 (1980).

    Google Scholar 

  72. S. Page, “Semihereditary and fully idempotent FPF rings,” Commun. Algebra, 11, No. 3, 227-242 (1983).

    Google Scholar 

  73. V. S. Ramamurthi, “On the injectivity and flatness of certain cyclic modules,” Proc. Amer. Math. Soc., 48, 21-25 (1975).

    Google Scholar 

  74. V. S. Ramamurthi and K. M. Rangaswamy, “Generalized V-rings,” Math. Scand., 31, 69-77 (1972).

    Google Scholar 

  75. G. Renault, “Sur les anneaux Atels que tout A-module a gauche non nul contient un sous-module maximal,” C. R. Acad. Sci. Paris, Ser. A-B264, A622-A624 (1967).

    Google Scholar 

  76. G. Renault, “Sur les anneaux Atels que tout A-module a gauche non nul contient un sous-module maximal,” C. R. Acad. Sci. Paris, Ser. A-B, 267, A792-A794 (1968).

    Google Scholar 

  77. S. H. Rim, “Semisimple Artinian localizations related with V-rings,” Commun. Korean Math. Soc., 10, No. 4, 839-847 (1995).

    Google Scholar 

  78. B. Sarath, “Krull dimension and Noetherianness,” Illinois J. Math., 20, No. 2, 329-353 (1976).

    Google Scholar 

  79. B. Sarath and K. Varadarajan, “Injectivity of certain classes of modules,” J. Pure Appl. Algebra, 5, 293-305 (1974).

    Google Scholar 

  80. J. Shapiro and M. Teply, “Semisimple localizations and V-rings,” Commun. Algebra, 16, No. 8, 1673-1688 (1988)

    Google Scholar 

  81. M. S. Shrikhande, “On hereditary and cohereditary modules,” Can. J. Math., 25, 892-896 (1973).

    Google Scholar 

  82. Y. Takehana, “V-rings relative to hereditary torsion theories,” Tsukuba J. Math., 6, No. 2, 293-298 (1982).

    Google Scholar 

  83. M. L. Teply, “On the idempotence and stability of kernel functors,” Glasgow Math. J., 37, No. 1, 37-43 (1995).

    Google Scholar 

  84. A. A. Tuganbaev, “Bass rings and perfect rings,” Usp. Mat. Nauk, 51, No. 1, 173-174 (1996).

    Google Scholar 

  85. A. A. Tuganbaev, “Rings over which each module has a maximal submodule,” Mat. Zametki, 61, No. 3, 407-415 (1997).

    Google Scholar 

  86. A. A. Tuganbaev, “Maximal submodules and locally perfect rings,” Mat. Zametki, 64, 136-142 (1998).

    Google Scholar 

  87. A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht (1998).

    Google Scholar 

  88. D. V. Tyukavkin, “Regular self-injective rings and V-rings,” Algebra Logika, 33, No. 5, 564-575 (1994).

    Google Scholar 

  89. K. Varadarajan, “Generalised V-rings and torsion theories,” Commun. Algebra, 14, No. 3, 455-467 (1986).

    Google Scholar 

  90. K. Varadarajan and K. Wehrhahn, “p-Injectivity of simple pre-torsion modules,” Glasgow Math. J.28, No. 2, 223-225 (1986).

    Google Scholar 

  91. D. G. Wang, “Generalized V-modules,” J. Qufu Normal Univ. Nat. Sci. Ed., 19, 30-34 (1993).

    Google Scholar 

  92. D. G. Wang, “On V-modules relative to torsion theory,” In: Rings and Radicals (Shijiazhuang, 1994), Pitman Res. Notes Math. Ser.Vol. 346, Longman, Harlow (1996), pp. 249-252.

    Google Scholar 

  93. D. G. Wang, “On some rings and self-injectivity,” Vietnam J. Math., 24, No. 2, 215-222 (1993).

    Google Scholar 

  94. R. Wisbauer, “Generalized cosemisimple modules,” Commun. Algebra, 18, No. 12, 4235-4253 (1990).

    Google Scholar 

  95. P. Wu, and X. Hu, “Some new characterizations of V-rings,” In.: Proc. 2nd Japan-China Int. Symp. Ring Theory and 28th Symp. Ring Theory (Okayama, 1995), Okayama Univ., Okayama (1996), pp. 147-149.

    Google Scholar 

  96. T. Wurfel, “Sur les V-anneaux semi-Artiniens,” Rev. Roumaine Math. Pures Appl., 20, No. 4, 503-507 (1975).

    Google Scholar 

  97. W. Xue, “Quasi-Hamsher modules and quasi-max rings,” Math. J. Okayama Univ., 39, 71-79 (1997).

    Google Scholar 

  98. W. Xue, “Characterizations of hereditary modules and V-modules,” Math. J. Okayama Univ., 39, 7-16 (1997).

    Google Scholar 

  99. W. Xue, “A note on Y J-injectivity,” Riv. Mat. Univ. Parma (6)1, 31-37 (1998).

    Google Scholar 

  100. M. F. Yousif, “V-modules with Krull dimension,” Bull. Austral. Math. Soc., 37, No. 2, 237-240 (1988).

    Google Scholar 

  101. R. Yue Chi Ming, “On generalizations of V-rings and regular rings,” Math. J. Okayama Univ., 20, No. 2, 123-129 (1978).

    Google Scholar 

  102. R. Yue Chi Ming, “On regular rings and V-rings,” Monatsh. Math., 88, No. 4, 335-344 (1979).

    Google Scholar 

  103. R. Yue Chi Ming, “On V-rings and prime rings,” J. Algebra, 62, No. 1, 13-20 (1980).

    Google Scholar 

  104. R. Yue Chi Ming, “On von Neumann regular rings. IV,” Riv. Mat. Univ. Parma, (4), 6, 47-54 (1981).

    Google Scholar 

  105. R. Yue Chi Ming, “On von Neumann regular rings. V,” Math. J. Okayama Univ., 22, No. 2, 151-160 (1980).

    Google Scholar 

  106. R. Yue Chi Ming, “On regular and continuous rings. II,” Kyungpook Math. J., 21, No. 2, 171-178 (1981).

    Google Scholar 

  107. R. Yue Chi Ming, “On V-rings and unit-regular rings,” Rend. Sem. Mat. Univ. Padova, 64, 127-140 (1981).

    Google Scholar 

  108. R. Yue Chi Ming, “Remarks on regular rings and V-rings,” Ricerche Mat., 30, No. 1, 3-14 (1981).

    Google Scholar 

  109. R. Yue Chi Ming, “On injective modules and annihilators,” Ricerche Mat., 33, No. 2, 147-158 (1984).

    Google Scholar 

  110. R. Yue Chi Ming, “On V-rings, p-V-rings and injectivity,” Kyungpook Math. J., 32, No. 2, 219-228 (1992).

    Google Scholar 

  111. R. Yue Chi Ming, “On self-injectivity and regularity,” Rend. Sem. Fac. Sci. Univ. Cagliari, No. 1, 9-24 (1994).

  112. R. Yue Chi Ming, “A note on YJ-injectivity,” Demonstr. Math., 30, No. 3, 551-556 (1997).

    Google Scholar 

  113. J. L. Zhang, “Characterizations of normal regular rings,” Northeast. Math. J., 10, No. 3, 359-364 (1994).

    Google Scholar 

  114. J. L. Zhang, “SF-rings whose maximal essential left ideals are ideals,” Adv. Math. (China), 23, No. 3, 257-262 (1994).

    Google Scholar 

  115. J. L. Zhang and X. N. Du, “Von Neumann regularity of SF-rings,” Commun. Algebra21, No. 7, 2445-2451 (1993).

    Google Scholar 

  116. S. H. Zhang, “T-Regular rings and TV-rings,” J. Math. Res. Expos., 11, No. 1, 31-36 (1991).

    Google Scholar 

  117. S. H. Zhang, “V-Modules and V-additive categories,” Northeast Math. J., 10, No. 3, 346-350 (1994).

    Google Scholar 

  118. M. Zhou, “The relation between commutative generalized regular rings and generalized V-rings,” Acta Math. Sinica, 35, No. 1, 140-143 (1992).

    Google Scholar 

  119. M. Zhou, “S-injective modules and S-Vrings,” Northeast Math. J., 8, No. 3, 303-310 (1992).

    Google Scholar 

  120. Y. Zhou, “Modules arising from some relative injectives,” Bull. Austral. Math. Soc., 53, No. 2, 249-260 (1996).

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tuganbaev, A.A. Rings Whose Nonzero Modules Have Maximal Submodules. Journal of Mathematical Sciences 109, 1589–1640 (2002). https://doi.org/10.1023/A:1013981125581

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013981125581

Keywords

Navigation