ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES

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

Derya Keskin Tütüncü Sultan Eylem Toksoy

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

Keywords

Absolute co-supplement (co-coclosed) module, Supplement (coclosed) submodule, 2000 AMS Classification: Primary 16 D 10. Secondary 06 C 05

ABSOLUTE CO-SUPPLEMENT AND ABSOLUTE CO-COCLOSED MODULES

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.