Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON SUBDIRECT PRODUCT OF PRIME MODULES

  • Received : 2016.05.04
  • Published : 2017.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

Keywords

References

  1. F. W. Anderson and K. R. Fuller, Rings and categories of modules, Granduate Text in Mathematics 13, Springer, Berline, 1973.
  2. O. D. Avraamova, A generalized density theorem, in Abelian groups and modules, No. 8 (Russian), 3-16, 172, Tomsk. Gos. Univ., Tomsk, 1989.
  3. M. Behboodi, A generalization of Baer's lower nilradical for modules, J. Algebra Appl. 6 (2007), no. 2, 337-353. https://doi.org/10.1142/S0219498807002284
  4. L. Bican, P. Jambor, T. Kepka, and P. Nemec, Prime and coprime modules, Fund. Math. 57 (1980), no. 1, 33-45.
  5. N. Dehghani and M. R. Vedadi, A characterization of modules embedding in products of primes and enveloping condition for their class, J. Algebra Appl. 14 (2015), no. 4, 1550051, 14 pp.
  6. E. Enochs, O. M. G. Jenda, and J. Xu, The existence of envelopes, Rend. Sem. Mat. Univ. Padova 90 (1993), 45-51.
  7. S. M. George, R. L. McCasland, and P. F. Smith, A principal ideal theorem analougue for modules over commutative rings, Comm. Algebra 22 (1994), no. 6, 2083-2099. https://doi.org/10.1080/00927879408824957
  8. A. Haghany and M. R. Vedadi, Endoprime modules, Acta Math. Hungar. 106 (2005), no. 1-2, 89-99. https://doi.org/10.1007/s10474-005-0008-2
  9. L. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15 (1963), 132-151. https://doi.org/10.4153/CJM-1963-016-1
  10. C. Lomp, Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions, J. Algebra Appl. 4 (2005), no. 1, 77-97. https://doi.org/10.1142/S0219498805001022
  11. J. C. McConnell and J. C. Robson, Non-commutative Noetherian Rings, Wiley-Interscience, New York, 1987.
  12. P. F. Smith and M. R. Vedadi, Essentially compressible modules and rings, J. Algebra 304 (2006), no. 2, 812-831. https://doi.org/10.1016/j.jalgebra.2005.08.018
  13. P. F. Smith and M. R. Vedadi, Submodules of direct sums of compressible modules, Comm. Algebra 36 (2008), no. 8, 3042-3049. https://doi.org/10.1080/00927870802110854
  14. Y. Tolooei and M. R. Vedadi, On rings whose modules have nonzero homomorphisms to nonzero submodules, Publ. Mat. 57 (2013), no. 1, 107-122. https://doi.org/10.5565/PUBLMAT_57113_04
  15. R. Wisbauer, Modules and Algebras: Bimodule Structure and Group Action on Algebras, Pitman Monograhs 81, Addison-Wesley-Longman, 1996.
  16. H. P. Yu, On quasiduo rings, Glasgow Math. J. 37 (1995), no. 1, 21-31. https://doi.org/10.1017/S0017089500030342
  17. J. Zelmanowitz, Weakly semisimple modules and density theory, Comm. Algebra 21 (1993), no. 5, 1785-1808. https://doi.org/10.1080/00927879308824653