Rings Whose Modules are ⊕ -Cofinitely Supplemented

Open access

Abstract

It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

Abstract

It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

References
  • 1. Alizade, R.; Bilhan, G.; Smith, P.F. - Modules whose maximal submodules have supplements, Comm. Algebra, 29 (2001), 2389-2405.

  • 2. Azumaya, G. - A characterization of semi-perfect rings and modules, Ring theory (Granville, OH, 1992), 28-40, World Sei. Publ., River Edge, NJ, 1993.

  • 3. Büyükaşik, E.; Lomp, С. - On a recent generalization of semiperfect rings, Bull. Aust. Math. Soc., 78 (2008), 317-325.

  • 4. Büyükaşik, E.; Demirci, Y.M. - Weakly distributive modules. Applications to supplement submodules, Proc. Indian Acad. Sei. Math. Sei., 120 (2010), 525-534.

  • 5. Çalişici, H.; Pancar, A. - <g>-cofinitely supplemented modules, Czechoslovak Math. J., 54 (2004), 1083-1088.

  • 6. Çalişici, H.; Pancar, A. - Confinitely semiperfect modules, Siberian Math. J., 46 (2005), 359-363.

  • 7. Gerasimov, V.N.; Sakhaev, 1.1. - A counterexample to two conjectures on projec- tive and flat modules, (Russian) Sibirsk. Mat. Zh., 25 (1984), 31-35.

  • 8. Idelhadj, A.; Tribak, R. - Modules for which every submodule has a supplement that is a direct summand, Arab. J. Sei. Eng. Sect. С Theme Issues, 25 (2000), 179-189.

  • 9. Idelhadj. A.; Tribak, R. - On some properties of Ф-supplemented modules, Int. J. Math. Math. Sei., 2003, 4373-4387.

  • 10. Kasch, F. - Modules and Rings, London Mathematical Society Monographs, 17, Academic Press, Inc., London-New York, 1982.

  • 11. Lomp, C. - On semilocal modules and rings, Comm. Algebra, 27 (1999), 1921-1935.

  • 12. Mohamed, S.H.; Müller, В.J. - Continuous and Discrete Modules, London Math- ematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990.

  • 13. Nisanci, B.; Pancar, A. - On generalization of ®-соfinitely supplemented modules, Ukrainian Math. J., 62 (2010), 203-209.

  • 14. Nakahara, S. - On a generalization of semiperfect modules, Osaka J. Math., 20 (1983), 43-50.

  • 15. Sharpe, D.W.; Vamos, P. - Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, 62, Cambridge University Press, London-New York, 1972.

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

  • 17. Wisbauer, R . - Foundations of Module and Ring Theory, A handbook for study and research. Algebra, Logic and Applications, 3. Gordon and Breach Science Publishers, Philadelphia, PA, 1991.

  • 18. Xue, W. - Characterizations of semiperfect and perfect rings, Publ. Mat., 40 (1996), 115-125.

1. Alizade, R.; Bilhan, G.; Smith, P.F. - Modules whose maximal submodules have supplements, Comm. Algebra, 29 (2001), 2389-2405.

2. Azumaya, G. - A characterization of semi-perfect rings and modules, Ring theory (Granville, OH, 1992), 28-40, World Sei. Publ., River Edge, NJ, 1993.

3. Büyükaşik, E.; Lomp, С. - On a recent generalization of semiperfect rings, Bull. Aust. Math. Soc., 78 (2008), 317-325.

4. Büyükaşik, E.; Demirci, Y.M. - Weakly distributive modules. Applications to supplement submodules, Proc. Indian Acad. Sei. Math. Sei., 120 (2010), 525-534.

5. Çalişici, H.; Pancar, A. - -cofinitely supplemented modules, Czechoslovak Math. J., 54 (2004), 1083-1088.

6. Çalişici, H.; Pancar, A. - Confinitely semiperfect modules, Siberian Math. J., 46 (2005), 359-363.

7. Gerasimov, V.N.; Sakhaev, 1.1. - A counterexample to two conjectures on projec- tive and flat modules, (Russian) Sibirsk. Mat. Zh., 25 (1984), 31-35.

8. Idelhadj, A.; Tribak, R. - Modules for which every submodule has a supplement that is a direct summand, Arab. J. Sei. Eng. Sect. С Theme Issues, 25 (2000), 179-189.

9. Idelhadj. A.; Tribak, R. - On some properties of Ф-supplemented modules, Int. J. Math. Math. Sei., 2003, 4373-4387.

10. Kasch, F. - Modules and Rings, London Mathematical Society Monographs, 17, Academic Press, Inc., London-New York, 1982.

11. Lomp, C. - On semilocal modules and rings, Comm. Algebra, 27 (1999), 1921-1935.

12. Mohamed, S.H.; Müller, В.J. - Continuous and Discrete Modules, London Math- ematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990.

13. Nisanci, B.; Pancar, A. - On generalization of ®-соfinitely supplemented modules, Ukrainian Math. J., 62 (2010), 203-209.

14. Nakahara, S. - On a generalization of semiperfect modules, Osaka J. Math., 20 (1983), 43-50.

15. Sharpe, D.W.; Vamos, P. - Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, 62, Cambridge University Press, London-New York, 1972.

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

17. Wisbauer, R . - Foundations of Module and Ring Theory, A handbook for study and research. Algebra, Logic and Applications, 3. Gordon and Breach Science Publishers, Philadelphia, PA, 1991.

18. Xue, W. - Characterizations of semiperfect and perfect rings, Publ. Mat., 40 (1996), 115-125.

Annals of the Alexandru Ioan Cuza University - Mathematics

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