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ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES

Year 2013, Volume: 42 Issue: 1, 67 - 79, 01.01.2013

Abstract

A module M is called an absolute co-coclosed (absolute co-supplement)module if whenever M ∼= T /X the submodule X of T is a coclosed (supplement) submodule of T .are absolute co-coclosed (absolute co-supplement) are precisely determined. We also investigate the rings whose (finitely generated) absolute co-supplement modules are projective. We show that a commutative domain R is a Dedekind domain if and only if every submodule of an absolute co-supplement R-module is absolute co-supplement.We also prove that the class Coclosed of all short exact sequences

References

  • Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules (New York: Springer, 19) Clark, J., Keskin T¨ut¨unc¨u D. and Tribak, R. Supplement submodules of injective modules, Comm. Algebra. 39, 4390–4402, 2011.
  • Clark, J., Lomp, C., Vanaja N. and Wisbauer, R. Lifting Modules, Supplements and Pro- jectivity in Module Theory (Frontiers in Math. Boston: Birkh¨auser, 2006.)
  • Dung, N. V., Hyunh, D. V., Smith P. F. and Wisbauer, R. Extending Modules, Pitman Research Notes in Mathematics Series (UK: Longman Scientific and Technical, 1994.) Erdo˘gan, (M. http://library.iyte.edu.tr/tezler/master/matematik/T000339.pdf. Institute Complement Modules Sc. ˙Izmir of Technology, 2004) Electronic copy:
  • Ganesan, L. and Vanaja, N. Modules for which every submodule has a unique coclosure, Comm. Algebra. 30 (5), 2355–2377, 2002.
  • Generalov, A. I. The ω-cohigh purity in a categories of modules, Math. Notes 33 (5-6) 402–408. Translated from Russian from Mat. Zametki 33 (5), 758–796, 1983.
  • Gerasimov, V. N. and Sakhaev, I. I. A counter example to two hypotheses on projective and flat modules, Sib. Mat. Zh. 25 (6), 31–35, 1984. English translation: Sib. Math. J. 24, 855–859, 1984.
  • Goodearl, K. R. Ring Theory: Nonsingular Rings and Modules (New York and Basel: Marcel Dekker Inc., 1976.)
  • Lam, T. Y. Lectures on Modules and Rings (New York, Berlin, Heidelberg: Springer, 1999.)
  • Mermut, E. Homological Approach to Complements and Supplements (Ph.D. dissertation, Dokuz Eyl¨ul University, 2004.)
  • Mishina A. P. and Skornyakov, L. A. Abelian groups and modules, American Mathematical Society Translations Series 2, 107, 1976. Translated from Russian from Abelevy gruppy i moduli, Izdat. Nauka, 1969.
  • Mohammed, A. and Sandomierski, F. L. Complements in projective modules, J. Algebra 127, 206–217, 1989.
  • Mohamed, S. H. and M¨uller, B. J. Continuous and Discrete Modules (London Math. Soc. Lecture Notes Series 147, Cambridge, 1990.)
  • Sklyarenko, E. G. Relative homological algebra in categories of modules, Russian Math. Surveys 33 (3), 97–137, 1978. Traslated from Russian from Uspehi Mat. Nauk 33 3(201), 85-120, 1978.
  • Talebi, Y. and Vanaja, N. The torsion theory cogenerated by M -small modules,Comm. Algebra 30 (3), 1449–1460, 2002.
  • Warfield, R. B. Jr. Serial rings and finitely presented modules, J. Algebra 37, 187–222, 1975.
  • Z¨oschinger, H. Projektive Moduln mit endlich erzeugtem Radikalfaktormodul, Math. Ann. 255 199–206, 1981.
  • Z¨oschinger, H. Schwach-injective moduln, Periodica Mathematica Hungarica 52 (2), 105– 128, 2006.

ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES

Year 2013, Volume: 42 Issue: 1, 67 - 79, 01.01.2013

Abstract

References

  • Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules (New York: Springer, 19) Clark, J., Keskin T¨ut¨unc¨u D. and Tribak, R. Supplement submodules of injective modules, Comm. Algebra. 39, 4390–4402, 2011.
  • Clark, J., Lomp, C., Vanaja N. and Wisbauer, R. Lifting Modules, Supplements and Pro- jectivity in Module Theory (Frontiers in Math. Boston: Birkh¨auser, 2006.)
  • Dung, N. V., Hyunh, D. V., Smith P. F. and Wisbauer, R. Extending Modules, Pitman Research Notes in Mathematics Series (UK: Longman Scientific and Technical, 1994.) Erdo˘gan, (M. http://library.iyte.edu.tr/tezler/master/matematik/T000339.pdf. Institute Complement Modules Sc. ˙Izmir of Technology, 2004) Electronic copy:
  • Ganesan, L. and Vanaja, N. Modules for which every submodule has a unique coclosure, Comm. Algebra. 30 (5), 2355–2377, 2002.
  • Generalov, A. I. The ω-cohigh purity in a categories of modules, Math. Notes 33 (5-6) 402–408. Translated from Russian from Mat. Zametki 33 (5), 758–796, 1983.
  • Gerasimov, V. N. and Sakhaev, I. I. A counter example to two hypotheses on projective and flat modules, Sib. Mat. Zh. 25 (6), 31–35, 1984. English translation: Sib. Math. J. 24, 855–859, 1984.
  • Goodearl, K. R. Ring Theory: Nonsingular Rings and Modules (New York and Basel: Marcel Dekker Inc., 1976.)
  • Lam, T. Y. Lectures on Modules and Rings (New York, Berlin, Heidelberg: Springer, 1999.)
  • Mermut, E. Homological Approach to Complements and Supplements (Ph.D. dissertation, Dokuz Eyl¨ul University, 2004.)
  • Mishina A. P. and Skornyakov, L. A. Abelian groups and modules, American Mathematical Society Translations Series 2, 107, 1976. Translated from Russian from Abelevy gruppy i moduli, Izdat. Nauka, 1969.
  • Mohammed, A. and Sandomierski, F. L. Complements in projective modules, J. Algebra 127, 206–217, 1989.
  • Mohamed, S. H. and M¨uller, B. J. Continuous and Discrete Modules (London Math. Soc. Lecture Notes Series 147, Cambridge, 1990.)
  • Sklyarenko, E. G. Relative homological algebra in categories of modules, Russian Math. Surveys 33 (3), 97–137, 1978. Traslated from Russian from Uspehi Mat. Nauk 33 3(201), 85-120, 1978.
  • Talebi, Y. and Vanaja, N. The torsion theory cogenerated by M -small modules,Comm. Algebra 30 (3), 1449–1460, 2002.
  • Warfield, R. B. Jr. Serial rings and finitely presented modules, J. Algebra 37, 187–222, 1975.
  • Z¨oschinger, H. Projektive Moduln mit endlich erzeugtem Radikalfaktormodul, Math. Ann. 255 199–206, 1981.
  • Z¨oschinger, H. Schwach-injective moduln, Periodica Mathematica Hungarica 52 (2), 105– 128, 2006.
There are 17 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Mathematics
Authors

Derya Keskin Tütüncü This is me

Sultan Eylem Toksoy This is me

Publication Date January 1, 2013
Published in Issue Year 2013 Volume: 42 Issue: 1

Cite

APA Tütüncü, D. K., & Toksoy, S. E. (2013). ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES. Hacettepe Journal of Mathematics and Statistics, 42(1), 67-79.
AMA Tütüncü DK, Toksoy SE. ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES. Hacettepe Journal of Mathematics and Statistics. January 2013;42(1):67-79.
Chicago Tütüncü, Derya Keskin, and Sultan Eylem Toksoy. “ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES”. Hacettepe Journal of Mathematics and Statistics 42, no. 1 (January 2013): 67-79.
EndNote Tütüncü DK, Toksoy SE (January 1, 2013) ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES. Hacettepe Journal of Mathematics and Statistics 42 1 67–79.
IEEE D. K. Tütüncü and S. E. Toksoy, “ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, pp. 67–79, 2013.
ISNAD Tütüncü, Derya Keskin - Toksoy, Sultan Eylem. “ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES”. Hacettepe Journal of Mathematics and Statistics 42/1 (January 2013), 67-79.
JAMA Tütüncü DK, Toksoy SE. ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES. Hacettepe Journal of Mathematics and Statistics. 2013;42:67–79.
MLA Tütüncü, Derya Keskin and Sultan Eylem Toksoy. “ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES”. Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, 2013, pp. 67-79.
Vancouver Tütüncü DK, Toksoy SE. ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES. Hacettepe Journal of Mathematics and Statistics. 2013;42(1):67-79.