Rings Whose Simple Singular Modules are PS-Injective

Xiang, Yueming;Ouyang, Lunqun

  • Received : 2012.11.13
  • Accepted : 2013.04.20
  • Published : 2014.09.23


Let R be a ring. A right R-module M is PS-injective if every R-homomorphism $f:aR{\rightarrow}M$ for every principally small right ideal aR can be extended to $R{\rightarrow}M$. We investigate, in this paper, rings whose simple singular modules are PS-injective. New characterizations of semiprimitive rings and semisimple Artinian rings are given.


PS-injective modules;semiprimitive rings;nonsingular rings


  1. K. R. Goodearl, Ring Theory II: Nonsingular rings and modules, (M. Dekker, New York, 1976).
  2. J. S. Alin and E. P. Armendariz, A class of rings having all singular simple modules injective, Math. Scand., 23(1968), 233-240.
  3. N. Q. Ding and J. L. Chen, Rings whose simple singular modules are Y J-injective, Math. Japonica, 40(1)(1994), 191-195.
  4. J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ., 32(1990), 73-76.
  5. Y. Hirano and H. Tominaga, Regular rings, V-rings and their generalizations, Hiroshima Math. J., 9(1979), 137-149.
  6. N. K. Kim, S. B. Nam and J. Y. Kim, On simple singular gp-injective modules, Comm. Algebra, 27(5)(1999), 2087-2096.
  7. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematic 189 (Springer-Verlag, 1999).
  8. G. Marks, On 2-primal Ore extensions, Comm. Algebra, 29(5)(2001), 2113-2123.
  9. S. B. Nam, N. K. Kim and J. Y. Kim, On simple GP-injective modules, Comm. Algebra, 23(1995), 5437-5444.
  10. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, (Cambridge University Press, Cambridge, 2003).
  11. M. B. Rege, On von Neumann regular rings and SF-rings, Math. Japon., 31(1986), 927-936.
  12. G. S. Xiao and W. T. Tong, Rings whose every simple left R-module is gp-injective, Southeast Asian Bull. Math., 30(2006), 969-980.
  13. Y. M. Xiang, Principally small injective rings, Kyungpook Math. J., 51(2)(2011), 177-185.
  14. R. Yue Chi Ming, On von Neumann regular rings(II), Math. Scand., 39(1976), 167-170.
  15. R. Yue Chi Ming, A note on biregular ring, Kyungpook, Math. J., 39(1999), 165-173.