On Generalized Flatness and Generalized SF-Rings


Abdullah M. Abdul-Jabbar

Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region, Iraq,



Abstract
The concept of generalized flatness and generalized SF-rings were first introduced by the
author in 2007. Now, in the present paper we continue to study generalized flatness,
generalized SF-rings and determine more properties of them. Moreover, several results
are proved. Finally, compare them with regular rings, left non-singular, strongly regular,
strongly π-regular, π-biregular and sπ-weakly regular rings under some conditions.
 

Key Words: SF-rings; generalized SF-rings; generalized flatness; regular rings 



References
[1] Abdul-Jabbar A. M., "Certain Types of Weak Regularity, GP-Injectivity and Flatness", Ph. D. Thesis,
College of Science, Salahaddin University-Erbil, (2007).
[2] Al-Kouri M. A., "On Strongly π-Regular Rings", Ph. D. Thesis, Mosul University, (1996).
[3] Aly F. S., "On right FGP-injective rings" International Math. Forum, 4(27); 1347-1355, (2009).
[4] Azumaya G., "Strongly π-regular rings" J. Fac. Sci. Hokkaido Univ., Vol.13; 34-39, (1954).
[5] Basheer D. A., "On N-Flat Modules", M. Sc. Thesis, Mosul University, (2011).
[6] Brown S. H., "Rings over which every simple modules is rationally complete" Canad J. Math.25; 693-
701, (1973).
[7] Burton D. M., "A First Course in Rings and Ideals", Addison Wesley Publishing Company, (1970).
[8] Goodearl K. R., "von Neumann Regular Rings", Pitman (London), (1979).
[9] Guangshi X. and Wenting T., "On GP-V-rings and characterizations of strongly regular rings" Northeast
Math. J., 18(4) ; 291-297, (2002).
[10] Gupta V., "Weakly π-regular rings and group rings" Mathematical Journal Okayama University, 19;
123-127, (1977).
[11] Habeb J. M., "A note on zero commutative and duo rings" Math. J. Okayama Univ., 32; 73-76, (1990).

[12] Jule Z., "Characterizations of strongly regular rings" Northeast Math. Journal, 10(3) ; 359-364, (1994).
[13] Mahmood R. D. and Husam Q. M., "On N-flat rings" Accepted in J. Al-Rafiden of Comp. and Math,
(2010).
[14] McCoy N. H., "Generalized regular rings" Bull. Amer. Math. Soc. 45; 175-178, (1939).
[15] Ming R. Y. C., "On von Neumann regular rings" Proc. Edinburgh Math. Soc. 19; 89-91, (1974).
[16] Ming R. Y. C., "On von Neumann regular rings III" Montash Math. 86; 251-257, (1978).
[17] Ming R. Y. C., "On injectivity and P-injectivity" J. Math. Kyoto University, 27; 439-452, (1978).
[18] Ming R. Y. C., "On regular rings and Artinian rings II" Riv. Math. Univ. Parma, 4(11) ; 101-109,
(1985).
[19] Ming R. Y. C., "On semi-prime and radical rings" Riv. Math. Univ. Parma, 4(12) ; 167-175, (1986).
[20] Ming R. Y. C., "On annihilator ideals (IV)" Riv. Math. Univ. Parma, (4)13; 19-27, (1987).
[21] Ming R. Y. C., "A note on regular rings III" Riv. Math. Univ. Parma, 6(1); 71-80, (1998).
[22] Naoum F. S., "On Semi-commutative π-Regular Rings", Ph. D. Thesis, Mosul University, (2004).
[23] Nicholson W. K. and Yousif M. F., "Quasi-Frobenius Rings", Cambridge Tracts In Mathematics,
Cambridge University Press, Cambridge, (2004).
[24] Shuker N. H., "On regularity and flatness" Raf. J. of Comp. & Math’s, 1(2); 11-16, (2004).
[25] Shuker N. H. and Yousif A. S., "A note on non-singular rings" Al-Rafiden J. of Comp. and Math’s,
2(2); 21-26, (2005).
[26] Subedi T., "Rings over which some classes of modules are flat: A survey", M. Sc. Thesis, North-Eastern
Hill University, (2010).
[27] von Neumann J., "On regular rings" Proc. Nat. Sci. U. S. A. 22; 707-713, (1936).
[28] Wei J. C., "Certain rings whose simple singular modules are nil-injective" Turk J. Math. 32; 393-408,
(2008).
[29] Yin X. B. , Wang R. and Li X. L., "The non-singularity and regularity of GP-V-rings" Journal of
Mathematical Research and Exposition, 31(6); 1003-1008, (2011).
[30] Yu H. P., "On quasi-duo ring" Glasgow Math. J., Vol. 37; 21-31, (1995).