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Simple continuous modules

Year 2019, Volume: 48 Issue: 5, 1336 - 1344, 08.10.2019

Abstract

A module $M$ is called a simple continuous module if it satisfies the conditions $(min-C_{1})$ and $(min-C_{2})$. A module $M$ is called singular simple-direct-injective if for any singular simple submodules $A$, $B$ of $M$ with $A\cong B\mid M$, then $A\mid M$. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple continuous modules and singular simple-direct-injective modules. For instance, it is shown that a ring $R$ is a right $V$-ring if and only if every right $R$-module is a simple continuous modules, and that a regular ring $R$ is a right $GV$-ring if and only if every cyclic right $R$-module is a singular simple-direct-injective module.

References

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Year 2019, Volume: 48 Issue: 5, 1336 - 1344, 08.10.2019

Abstract

References

  • [1] I. Amin, Y. Ibrahim and M.F. Yousif, C3-modules, Algebra Colloq. 22, 655-670, 2015.
  • [2] I. Amin, M.F. Yousif and N. Zeyada, Soc-injective rings and modules, Comm. Algebra 33, 4229-4250, 2005.
  • [3] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin, New York, 1974.
  • [4] J.-E. Björk, Rings satisfying certain chain conditions, J. Reine Angew. Math. 245, 63-73, 1970.
  • [5] V. Camillo, Y. Ibrahim, M.F. Yousif and Y.Q. Zhou, Simple-direct-injective modules, J. Algebra 420, 39-53, 2014.
  • [6] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Birkhäuser Basel, 2006.
  • [7] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending Modules, Longman Scientific and Technical, 1994.
  • [8] C. Faith, Algebra II: Ring Theory, Springer-Verlag, Berlin, New York, 1976.
  • [9] J.W. Fisher, Von Neumann regular rings versus V-rings, in: Lect. Notes Pure Appl. Math. 7, 101-119, Dekker, New York, 1974.
  • [10] F. Kasch, Modules and Rings, London Math. Soc. Monogr. 17, Academic Press, New York, 1982.
  • [11] S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, UK, 1990.
  • [12] W.K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math. 158, Cambridge Univ. Press, Cambridge, UK, 2003.
  • [13] A.C. Özcan, A. Harmanci and P.F. Smith, Duo modules, Glasgow Math. J. 48, 533- 545, 2006.
  • [14] V.S. Ramamurthi and K.M. Rangaswamy, Generalized V -rings, Math. Scand. 31, 69-77, 1972.
  • [15] P.F. Smith, CS-modules and Weak CS-modules, Non-commutative Ring Theory, 99- 115, Springer LNM 1448, 1990.
  • [16] P.F. Smith, Modules for which every submodule has a unique closure, in: Ring Theory, 303-313, World Scientific, Singapore, 1993.
  • [17] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadel- phia, 1991.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Yongduo Wang This is me 0000-0002-0756-3899

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Wang, Y. (2019). Simple continuous modules. Hacettepe Journal of Mathematics and Statistics, 48(5), 1336-1344.
AMA Wang Y. Simple continuous modules. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1336-1344.
Chicago Wang, Yongduo. “Simple Continuous Modules”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1336-44.
EndNote Wang Y (October 1, 2019) Simple continuous modules. Hacettepe Journal of Mathematics and Statistics 48 5 1336–1344.
IEEE Y. Wang, “Simple continuous modules”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1336–1344, 2019.
ISNAD Wang, Yongduo. “Simple Continuous Modules”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1336-1344.
JAMA Wang Y. Simple continuous modules. Hacettepe Journal of Mathematics and Statistics. 2019;48:1336–1344.
MLA Wang, Yongduo. “Simple Continuous Modules”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1336-44.
Vancouver Wang Y. Simple continuous modules. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1336-44.