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ON SOME GENERALIZATIONS OF CLOSED SUBMODULES

  • Received : 2014.08.13
  • Published : 2015.09.30

Abstract

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially dierent ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.

References

  1. E. Buyukasik and Y. Durgun, Coneat submodules and coneat-flat modules, J. Korean Math. Soc. 51 (2014), no. 6, 1305-1319. https://doi.org/10.4134/JKMS.2014.51.6.1305
  2. I. Crivei, s-pure submodules, Int. J. Math. Math. Sci. 4 (2005), no. 4, 491-497.
  3. S. Crivei, m-injective modules, Mathematica 40(63) (1998), no. 1, 71-78.
  4. S. Crivei, Neat and coneat submodules of modules over commutative rings, Bull. Aust. Math. Soc. 89 (2014), no. 2, 343-352. https://doi.org/10.1017/S0004972713000622
  5. N. V. Dung, D. V. Huynh, P. F. Smith, and R. Wisbauer, Extending Modules, Longman Scientific & Technical, Harlow, 1994.
  6. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter & Co., Berlin, 2000.
  7. L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64 (2012), no. 2, 131-143. https://doi.org/10.1007/s10998-012-7509-x
  8. A. I. Generalov, Weak and ${\omega}$-high purities in the category of modules, Mat. Sb. (N.S.) 105(147) (1978), no. 3, 389-402, 463.
  9. K. Honda, Realism in the theory of abelian groups. I, Comment. Math. Univ. St. Paul. 5 (1956), 37-75.
  10. T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1999.
  11. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, American Mathematical Society, Providence, RI, 2001.
  12. C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc. 26 (1970), 561-566. https://doi.org/10.1090/S0002-9939-1970-0294409-8
  13. E. Mermut, C. Santa-Clara, P. F. Smith, Injectivity relative to closed submodules, J. Algebra 321 (2009), no. 2, 548-557. https://doi.org/10.1016/j.jalgebra.2008.11.004
  14. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius rings, Cambridge University Press, Cambridge, 2003.
  15. K. Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), no. 3, 310-338. https://doi.org/10.14492/hokmj/1381757705
  16. G. Renault, Etude de certains anneaux lies aux sous-modules complements d'un A-module, C. R. Acad. Sci. Paris 258 (1964), 4888-4890.
  17. L. Salce, Almost perfect domains and their modules, Commutative algebraNoetherian and non-Noetherian perspectives, 363-386, Springer, New York, 2011.
  18. C. Santa-Clara and P. F. Smith, Modules which are self-injective relative to closed submodules, Algebra and its applications (Athens, OH, 1999), 487-499, Contemp. Math., 259, Amer. Math. Soc., Providence, RI, 2000.
  19. C. Santa-Clara and P. F. Smith, Direct products of simple modules over Dedekind domains, Arch. Math. 82 (2004), no. 1, 8-12. https://doi.org/10.1007/s00013-003-0840-y
  20. E. G. Sklyarenko, Relative homological algebra in the category of modules, Uspehi Mat. Nauk 33 (1978), no. 3, 85-120.
  21. P. F. Smith, Injective modules and prime ideals, Comm. Algebra 9 (1981), no. 9, 989-999. https://doi.org/10.1080/00927878108822627
  22. B. T. Stenstrom, High submodules and purity, Ark. Mat. 7 (1967), 173-176. https://doi.org/10.1007/BF02591033
  23. A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht, 1998.
  24. H. Zoschinger, Gelfandringe und koabgeschlossene Untermoduln, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. (1982), 43-70.
  25. H. Zoschinger, Swach-Flache Moduln, Comm. Algebra 41 (2013), no. 12, 4393-4407. https://doi.org/10.1080/00927872.2012.699570