대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제56권2호
  • /
  • Pages.505-514
  • /
  • 2019
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON MONOFORM MODULES

  • 투고 : 2018.04.19
  • 심사 : 2018.12.10
  • 발행 : 2019.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let R be a commutative ring with identity and M be a unitary R-module. A submodule N of M is called a dense submodule if $Hom_R(M/N,\;E_R(M))=0$, where $E_R(M)$ is the injective hull of M. The R-module M is said to be monoform if any nonzero submodule of M is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform. (1) The prime R-module M such that for any nonzero submodule N of M, $Ann_R(M/N){\neq}Ann_R(M)$. (2) Strongly prime R-module. (3) Faithful multiplication module over an integral domain.

키워드

참고문헌

  1. M. M. Ali, Residual submodules of multiplication modules, Beitrage Algebra Geom. 46 (2005), no. 2, 405-422.
  2. B. Amini, M. Ershad, and H. Sharif, Coretractable modules, J. Aust. Math. Soc. 86 (2009), no. 3, 289-304. https://doi.org/10.1017/S1446788708000360
  3. N. Bourbaki, Commutative algebra. Chapters 1-7, translated from the French, reprint of the 1989 English translation, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.
  4. J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156-181.
  5. Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra 16 (1988), no. 4, 755-779. https://doi.org/10.1080/00927878808823601
  6. E. H. Feller and E. W. Swokowski, Prime modules, Canad. J. Math. 17 (1965), 1041-1052. https://doi.org/10.4153/CJM-1965-099-5
  7. G. D. Findlay and J. Lambek, A generalized ring of quotients. I, II, Canad. Math. Bull. 1 (1958), 77-85, 155-167. https://doi.org/10.4153/CMB-1958-009-3
  8. R. Gordon and J. C. Robson, Krull dimension, American Mathematical Society, Providence, RI, 1973.
  9. A. Hajikarimi, Local cohomology modules which are supported only at finitely many maximal ideals, J. Korean Math. Soc. 47 (2010), no. 3, 633-643. https://doi.org/10.4134/JKMS.2010.47.3.633
  10. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
  11. H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid, second edition, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989.
  12. A. S. Mijbass, Quasi-Dedekind modules, Ph.D. Thesis, College of Science University of Baghdad, 1997.
  13. A. R. Naghipour, Strongly prime submodules, Comm. Algebra 37 (2009), no. 7, 2193-2199. https://doi.org/10.1080/00927870802467239
  14. A. R. Naghipour, Some results on strongly prime submodules, J. Algebr. Syst. 1 (2013), no. 2, 79-89.
  15. D. E. Rush, Strongly prime submodules, G-submodules and Jacobson modules, Comm. Algebra 40 (2012), no. 4, 1363-1368. https://doi.org/10.1080/00927872.2010.551530
  16. R. Y. Sharp, Steps in commutative algebra, second edition, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
  17. S. Singh and Y. Al-Shaniafi, Multiplication modules, Comm. Algebra 29 (2001), no. 6, 2597-2609. https://doi.org/10.1081/AGB-100002410
  18. J. M. Zelmanowitz, Representation of rings with faithful polyform modules, Comm. Algebra 14 (1986), no. 6, 1141-1169. https://doi.org/10.1080/00927878608823357