Aspects of Weak, $s$-cs and Almost Injective Rings

Authors

  • Nasr A. Zeyada Cairo University, Faculty of Science Department of Mathematics, Egypt; University of Jeddah, Faculty of Science, Department of Mathematics, Saudi Arabia
  • Amr K. Amin Beni-Suef University, Faculty of Science, Departement of Mathematics, Egypt; Umm Al-Qura University, Jamoum University College, Department of Mathematics, Makkah, Saudi Arabia

DOI:

https://doi.org/10.5644/SJM.12.2.05

Keywords:

$CS$, weak-$CS$, $S$-$CS$, rad-injective rings, almost injective rings, Kasch rings, Quasi-Frobenius rings

Abstract

It is not known whether right $CF$-rings ($FGF$-rings) are right artinian (quasi-Frobenius). This paper gives a positive answer of this question in the case of weak $CS$ ($s$-$CS$) and $GC2$ rings. Also we get some new results on almost injective rings.

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References

I. Amin, M. Yousif and N. Zeyada, Soc-injective rings and modules, Comm. Algebra, 33 (2005), 4229–4250.

F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin-New York, 1974.

J. E. Bjork, Rings satisfying certain chain conditions, J. Reine Angew. Math., 245 (1970), 63–73.

V. Camillo, W. K. Nicholson and M. F. Yousif, Ikeda-Nakayama rings, J. Algebra, 226 (2000), 1001–1010.

N. V. Dung, V. D. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman Reasearch Notes in Math. Longman, 1994.

C. Faith, Rings with ascending chain conditions on annihilators, Nagoya Math. J., 27 (1966), 179–191.

C. Faith, Algebra II, Ring Theory, Springer-Verlag, Berlin-New York, 1976.

J. L. G´omez Pardo and P. A. Guil Asensio, Essential embedding of cyclic modules in projectives, Trans. Amer. Math. Soc., 349 (1997), 4343–4353.

J. L. G´omez Pardo and P. A. Guil Asensio, Rings with finite essential socle, Proc. Amer. Math. Soc., 125 (1997), 971–977.

J. L. G´omez Pardo and M. F. Yousif, Semiperfect min-CS rings, Glasg. Math.J., 41 (1999), 231–238.

Dinh Van Huynh, S. K. Jain and S. R. Lopez-Permouth, When cyclic singular modules over a simple ring are injective, J. Algebra, 263 (2003), 188–192.

F. Kasch, Modules and Rings, Academic Press, New York, 1982.

C. Lomp, On semilocal modules and rings, Comm. Algebra, 27 (4) (1999), 1921–1935.

G. O. Michler and O.E. Villamayor, On rings whose simple modules are injective, J. Algebra, 25 (1973), 185–201.

S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, Cambridge University Press, Cambridge, 1990.

W. K. Nicholson and M.F. Yousif, Mininjective rings, J. Algebra, 187 (1997), 548–578.

W. K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math., 158, Cambridge University Press, Cambridge, 2003.

B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra, 4 (1966), 373–387.

Liang Shen and Jianlong Chen, New characterizations of quasi-Frobenius rings, Comm. Algebra, 34 (2006), 2157–2165.

P. F. Smith, CS -modules and weak CS -modules, Noncommutative Ring Theory, Springer LNM, 1448 (1990), 99–115.

P. Vamos, The dual of the notion of ‘finitely generated’, J. London Math. Soc., 43 (1968), 186–209.

M. F. Yousif, Y. Zhou and N. Zeyada, On pseudo-Frobenius rings, Canad. Math. Bull., 48 (2) (2005), 317–320.

N. Zeyada, S. Hussein and A. Amin, Rad-injective and almost-injective modules and rings, Algebra Colloq., 18 (3) (2011), 411–418.

N. Zeyada and N. Jarboui, s-CS modules and rings, Int. J. Algebra, 7 (2) 2013, 49–62

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Published

30.05.2024

How to Cite

Zeyada, N. A., & Amin, A. K. (2024). Aspects of Weak, $s$-cs and Almost Injective Rings. Sarajevo Journal of Mathematics, 12(2), 183–192. https://doi.org/10.5644/SJM.12.2.05

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Articles