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On Generalizations of Extending Modules

  • Karabacak, Fatih (Anadolu University, Education Faculty, Department of Mathematics)
  • Received : 2008.08.03
  • Accepted : 2008.11.04
  • Published : 2009.09.30

Abstract

A module M is said to be SIP-extending if the intersection of every pair of direct summands is essential in a direct summand of M. SIP-extending modules are a proper generalization of both SIP-modules and extending modules. Every direct summand of an SIP-module is an SIP-module just as a direct summand of an extending module is extending. While it is known that a direct sum of SIP-extending modules is not necessarily SIP-extending, the question about direct summands of an SIP-extending module to be SIP-extending remains open. In this study, we show that a direct summand of an SIP-extending module inherits this property under some conditions. Some related results are included about $C_{11}$ and SIP-modules.

Keywords

SIP-extending modules;summand intersection property;extending modules

References

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, 1974.
  2. D. M. Arnold and J. Hausen, A characterization of modules with the summand intersection property, Comm. Algebra, 18(1990), 519-528. https://doi.org/10.1080/00927879008823929
  3. G. F. Birkenmeier, F. Karabacak and A. Tercan, When is the SIP (SSP) property inherited by free modules, Acta Math. Hungar., 112(2006), 103-106. https://doi.org/10.1007/s10474-006-0067-z
  4. G. F. Birkenmeier, J. Y. Kim and J. K. Park, When is the CS condition hereditary?, Comm. Algebra, 27(1999), 3785-3885. https://doi.org/10.1080/00927879908826670
  5. N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman, 1990.
  6. J. Hausen, Modules with the summand intersection property, Comm. Algebra, 17(1989), 135-148. https://doi.org/10.1080/00927878908823718
  7. I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, 1969.
  8. F. Karabacak and A. Tercan, Matrix rings with the summand intersection property, Czech. Math. J., 53(2003), 621-626. https://doi.org/10.1023/B:CMAJ.0000024507.03939.ce
  9. F. Karabacak and A. Tercan, On modules and matrix rings with SIP-extending, Taiwanese J. Math., 11(2007), 1037-1044. https://doi.org/10.11650/twjm/1500404800
  10. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, Journal of Algebra, 223(2000), 477-488. https://doi.org/10.1006/jabr.1999.8017
  11. S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, Cambridge University Press, 1990.
  12. K. Oshiro and S. T. Rizvi, Exchange property of quasi-continuous modules with the finite exchange property, Osaka J. Math., 33(1996), 217-234.
  13. S. T. Rizvi and C. S. Roman, Baer and quasi-baer modules, Comm. Algebra, 32(2004), 103-123. https://doi.org/10.1081/AGB-120027854
  14. P. F. Smith, Modules for which every submodule has a unique closure, Ring Theory (S.Jain and S.T. Rizvi eds.) New Jersey, World Scientific, (1992), 302-317.
  15. P. F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra, 21(1993), 1809-1847. https://doi.org/10.1080/00927879308824655
  16. P. F. Smith and A. Tercan, Direct summands of modules which satisfy (C11), Algebra Colloq., 11(2004), 231-237.
  17. G. V. Wilson, Modules with the summand intersection property, Comm. Algebra, 14(1986), 21-38. https://doi.org/10.1080/00927878608823297

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  1. CS-Rickart modules vol.35, pp.4, 2014, https://doi.org/10.1134/S199508021404009X