Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE

  • Lee, Sang Cheol (Department of Mathematics Education, and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • Song, Yeong Moo (Department of Mathematics Education, Sunchon National University) ;
  • Varmazyar, Rezvan (Department of Mathematics, Khoy Branch, Islamic Azad University)
  • Received : 2017.04.03
  • Accepted : 2017.05.10
  • Published : 2017.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

Keywords

References

  1. Reza Ameri, On the prime submodules of multiplication modules, Int. J. Math. Math. Sci. (2003), no. 27, 1715-1724.
  2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  3. Kamran Divaani-Aazar and Mohammad Ali Esmkhani, Associated prime submodules of finitely generated modules, Comm. Algebra 33 (2005), no. 11, 4259-4266. https://doi.org/10.1080/00927870500279027
  4. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, With a view toward algebraic geometry.
  5. Zeinab Abd El-Bast and Patrick F. Smith, Multiplication modules, Comm. Al-gebra 16 (1988), no. 4, 755-779. https://doi.org/10.1080/00927878808823601
  6. Friedrich Ischebeck and Ravi A. Rao, Ideals and reality, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005, Projective modules and number of generators of ideals.
  7. Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970.
  8. Max. D. Larsen and Paul J. McCarthy, Multiplicative theory of ideals, Academic Press, New York-London, 1971, Pure and Applied Mathematics, Vol. 43.
  9. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986, Translated from the Japanese by M. Reid.
  10. R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math. 23 (1993), no. 3, 1041-1062. https://doi.org/10.1216/rmjm/1181072540
  11. Fazal Mehdi, On multiplication modules, Math. Student 42 (1974), 149-153 (1975).
  12. Hosein Fazaeli Moghimi and Javad Bagheri Harehdashti, Mappings between lattices of radical submodules, Int. Electron. J. Algebra 19 (2016), 35-48. https://doi.org/10.24330/ieja.266191
  13. R. Y. Sharp, Steps in commutative algebra, London Mathematical Society Student Texts, vol. 19, Cambridge University Press, Cambridge, 1990.
  14. D. W. Sharpe and P. Vamos, Injective modules, Cambridge University Press, London-New York, 1972, Cambridge Tracts in Mathematics and Mathematical Physics, No. 62.
  15. Patrick F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel) 50 (1988), no. 3, 223-235. https://doi.org/10.1007/BF01187738
  16. Patrick F. Smith, Mappings between module lattices, Int. Electron. J. Algebra 15 (2014), 173-195. https://doi.org/10.24330/ieja.266246
  17. Patrick F. Smith, Fully invariant multiplication modules, Palest. J. Math. 4 (2015), no. Special issue, 462-470.