SOME CHARACTERIZATIONS OF EF-EXTENDING RINGS

Abstract

Thuyet and Wisbauer considered the extending property for the class of (essentially) finitely generated submodules. A module M is called ef-extending if every closed submodule which contains essentially a finitely generated submodule is a direct summand of M. A ring R is called right ef-extending if RR is an ef-extending module. We show that a ring R is QF if and only if R is a left Noetherian, right GP-injective and right efextending ring. Moreover, we prove that R is right PF if and only if R is a right cogenerator, right ef-extending and I-finite.

Keywords

• ef-extending rings, extending (or CS) rings, PF rings, QF rings

References

1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1974.
2. J. Chen and N. Ding, On general principally injective rings, Comm. Algebra, (5)(1999), 2097 - 2116.
3. J. L. Chen and W. X. Li, On artiness of right CF rings, Comm. Algebra, (11) (2004), 4485 - 4494.
4. J. Chen, L. Shen and Y. Zhou, Characterization of QF rings, Comm. Algebra, (2007), 281-288.
5. N. Chien and L. V. Thuyet, On ef-extending modules, Southeast Asian Bull. Math., 26 (2003), 909-916.
6. N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman, 1996.
7. C. Faith, Algebra II: Ring Theory, Springer-Verlag, Berlin, (1976).
8. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Dekker, New York, 1976.

9. J. L. G´omez Pardo and P. A. Guil Asensio, Rings with finite essential socle, Proc. Amer. Math. Soc., 125 (1997), 971-977.
10. A. Harmanci and P. F. Smith, Finite direct sums of CS-Modules, Houston J. Math., 19(4)( 1995), 523-532.
11. F. Kasch, Modules and Rings, Academic Press, London, New York, 1982.
12. T. Y. Lam, A First Course in Noncommutative Rings, Springer Graduate Text, S. H. Mohammed and B. J. M¨uller, Continous and Discrete Modules, London Math. Soc., LN 147, Cambridge Univ. Press, 1990.
13. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Univ. Press, 2003.
14. W. K. Nicholson and M. F. Yousif, CS rings and Nakayama permutations, Comm. Algebra, 25(1997), 3787-3795.
15. T. C. Quynh and L. V. Thuyet, On rings with ACC on annihilators and having essential socles, Proc. Int. Math. and Appl. (ICMA, Bangkok 2005), Contri- bution in Math. and Appl., 227-234, 2006.
16. T. C. Quynh and L. V. Thuyet, Some properties of ef-extending rings, to appear in Math. J. Okayama Univ. L. V. Thuyet and T. C. Quynh, On general injective rings with chain condi- tions, to appear in Algebra Coll. L. V. Thuyet and R. Wisbauer, Extending property for finitely generated sub- modules, Vietnam J. Math., 25 (1997), 65 - 73.
17. R. Wisbauer, Foundations of Module and Ring Theory; Gordon and Breach, Reading, 1991.
18. M. F. Yousif and Y. Zhou, Pseudo-Frobenius rings: characterizations and ques- tions, Comm. Algebra, 31(9)(2003), 4473 - 4484.
19. Y. Zhou, Rings in which certain right ideals are direct summands of annihila- tors, J. Aust. Math. Soc., 73 (2002), 335 - 346. Truong Cong Quynh
20. Department of Mathematics Danang University Ton Duc Thang DaNang city, Vietnam e-mails: [e-mail] [e-mail]