• Arabi-Kakavand, Marzieh (Department of Mathematical Sciences Isfahan University of Technology) ;
  • Asgari, Shadi (Department of Mathematical Sciences University of Isfahan) ;
  • Tolooei, Yaser (Department of Mathematics Faculty of Science Razi University)
  • Received : 2016.03.03
  • Published : 2017.03.31


We investigate modules M having the injective property relative to nonsingular modules. Such modules are called "$\mathcal{N}$-injective modules". It is shown that M is an $\mathcal{N}$-injective R-module if and only if the annihilator of $Z_2(R_R)$ in M is equal to the annihilator of $Z_2(R_R)$ in E(M). Every $\mathcal{N}$-injective R-module is injective precisely when R is a right nonsingular ring. We prove that the endomorphism ring of an $\mathcal{N}$-injective module has a von Neumann regular factor ring. Every (finitely generated, cyclic, free) R-module is $\mathcal{N}$-injective, if and only if $R^{(\mathbb{N})}$ is $\mathcal{N}$-injective, if and only if R is right t-semisimple. The $\mathcal{N}$-injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the $\mathcal{N}$-injective property, we determine the rings whose all nonsingular cyclic modules are injective.


Supported by : IPM


  1. F.W. Anderson and K. R. Fuller, Rings and Categories of Modules, New York, Springer-Verlag, 1992.
  2. Sh. Asgari and A. Haghany, t-Extending modules and t-Baer modules, Comm. Algebra 39 (2011), no. 5, 1605-1623.
  3. Sh. Asgari, A. Haghany, and Y. Tolooei, t-Semisimple modules and t-semisimple rings, Comm. Algebra 41 (2013), no. 5, 1882-1902.
  4. N. V. Dung, D. V. Huynh, P. F. Smith, and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics 313, Harlow, Longman, 1994.
  5. C. Faith, When are proper cyclics injective?, Pacific J. Math. 45 (1973), 97-112.
  6. K. R. Goodearl, Singular Torsion and Splitting Properties, Mem. Amer. Math. Soc. No. 124, AMS, 1972.
  7. K. R. Goodearl and R. B. Warfield Jr., An Introduction to Noncommutative Noether-ian Rings, 2nd ed. London Mathematical Society Student Texts, Vol. 16. Cambridge: Cambridge University Press, 2004.
  8. S. K. Jain, A. K. Srivastava, and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford University Press, 2012.
  9. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189, Berlin, New York: Springer-Verlag, 1998.
  10. G. Lee, S. T. Rizvi, and C. S. Roman, Modules whose endomorphism rings are von Neumann regular, Comm. Algebra 41 (2013), no. 11, 4066-4088.
  11. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math-ematics, Vol. 158, Cambridge: Cambridge University Press, 2003.
  12. B. L. Osofsky, Homological properties of rings and modules, Rutgers University, Doctoral Dissertation, 1964.
  13. B. L. Osofsky and P. F. Smith, Cyclic modules whose quotients have all complements submodules direct summands, J. Algebra 139 (1991), no. 2, 342-354.
  14. S. T. Rizvi and M. F. Yousif, On continuous and singular modules, Noncommutative Ring Theory, Proc., Athens, Lecture Notes in Mathematics, Vol. 1448, pp. 116-124, Berlin, New York and Heidelberg: Springer Verlag, 1990.
  15. M. F. Yousif, Y. Zhou, and N. Zeyad, On pseudo-Frobenius rings, Canad. Math. Bull. 48 (2005), no. 2, 317-320.

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