Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

When Some Complement of an EC-Submodule is a Direct Summand

  • Received : 2009.03.25
  • Accepted : 2009.10.27
  • Published : 2010.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

A module M is said to satisfy the $EC_{11}$ condition if every ec-submodule of M has a complement which is a direct summand. We show that for a multiplication module over a commutative ring the $EC_{11}$ and P-extending conditions are equivalent. It is shown that the $EC_{11}$ property is not inherited by direct summands. Moreover, we prove that if M is an $EC_{11}$-module where SocM is an ec-submodule, then it is a direct sum of a module with essential socle and a module with zero socle. An example is given to show that the reverse of the last result does not hold.

Keywords

References

  1. M. M. Ali, D. J. Smith, Pure submodules of multiplication modules, Beitrage zur Algebra und Geometrie, Vol. 45(2004), 61-74.
  2. F.W. Anderson, K.R. Fuller, Rings and Catogories of Modules, Springer-Verlag, New York, 1992.
  3. G.F. Birkenmeier, A. Tercan, When some complement of a submodule is a summand, Comm. Algebra, 35(2)(2007), 597-611. https://doi.org/10.1080/00927870601074830
  4. G.F. Birkenmeier, B.J. Muller, S.T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra, 30(3)(2002), 1395-1415. https://doi.org/10.1080/00927870209342387
  5. G.F. Birkenmeier, J.Y. Kim, J.K. Park, When is the CS condition hereditary?, Comm. Algebra, 27(8)(1999), 3875-3885. https://doi.org/10.1080/00927879908826670
  6. C. Celep Yucel and A. Tercan, Modules whose ec-closed submodules are direct summand, Taiwanese J. Math., 13(2009), 1247-1256. https://doi.org/10.11650/twjm/1500405505
  7. N.V. Dung, D.V. Huynh, P.F. Smith, R. Wisbauer, Extending Modules, Longman Scientific and Technical, Harlow, Essex, England, 1994.
  8. Z. A. El-Bast, P. F. Smith, Multiplication modules, Comm. Algebra, 16(1988), 755-779. https://doi.org/10.1080/00927878808823601
  9. M.A. Kamal, O.A. Elmnophy, On P-extending Modules, Acta. Math. Univ. Comenianae, 74 ,(2005), 279-286.
  10. T.Y. Lam, Lectures on Modules and Rings, Springer, New York, 1999.
  11. P.F. Smith, A. Tercan, Generalizations of CS-Modules, Comm. Algebra, 21(6)(1993), 1809-1847. https://doi.org/10.1080/00927879308824655
  12. P.F. Smith, A. Tercan, Direct summands of modules which satisfy ($C_{11}$), Algebra Colloq., 11(2004), 231-237.
  13. B. Stenstrom, Rings of Quotients, Springer-Verlag, New York, 1975.
  14. F. Takıl, A. Tercan, Modules whose submodules essentially embedded in direct summands, Comm. Algebra, 37(2009), 460-469. https://doi.org/10.1080/00927870802248688
  15. G.V. Wilson, Modules with summand intersection property, Comm. Algebra, 14(1)(1986), 21-38. https://doi.org/10.1080/00927878608823297