On SF-rings and Regular Rings

• Journal title : Kyungpook mathematical journal
• Volume 53, Issue 3,  2013, pp.397-406
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2013.53.3.397
Title & Authors
On SF-rings and Regular Rings
Subedi, Tikaram; Buhphang, Ardeline Mary;

Abstract
A ring R is called a left (right) SF-ring if simple left (right) R-modules are flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if R is a left SF-ring such that $\small{l(x)(r(x))}$ is an essential left (right) ideal for every right (left) zero divisor x of R, then R is a division ring.
Keywords
Left SF-rings;von Neumann regular rings;strongly regular rings;weakly left ideals;W-ideals;
Language
English
Cited by
References
1.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules (Graduate Text in Mathematics 13), Springer -Verlag, Berlin Heldelberg-New York, (1974).

2.
V. S. Ramamurthy, Weakly regular rings, Canad. Math. Bull., 16(3)(1973), 317-321.

3.
V. S. Ramamurthy, On the injectivity and flatness of certain cyclic modules, Proc. Amer. Math. Soc., 48(1975), 21-25.

4.
M. B. Rege, On von Neumann regular rings and SF-rings, Math. Japonica, 31(6)(1986), 927-936.

5.
X. Song and X. Yin, Generalizations of V-rings, Kyungpook Math. J., 45(2005), 357-362.

6.
B. T. Strenstrom, Rings of quotients- An introduction to the methods of ring theory, Springer-Verlag, New York, (1975).

7.
G. Xiao, On GP-V-rings and characterizations of strongly regular rings, Northeast. Math. J., 18(4)(2002), 291-297.

8.
Y. Xiao, One sided SF-rings with certain chain conditions, Canad. Math. Bull., 37(2)(1994), 272-277.

9.
R. Yue. Chi. Ming, On von Neumann regular rings, Proc. Edinburgh Math. Soc., 19(1974), 89-91.

10.
R. Yue. Chi. Ming, On von Neumann regular rings, VIII, J. Korean Math. Soc., 19 (2)(1983), 97-104.

11.
J. Zhang, Characterizations of strongly regular rings, Northeast. Math. J., 10(3)(1994), 359-364.

12.
J. Zhang and X. Du, Von Neumann regularity of SF-rings, Communications in Algebra, 21(7)(1993), 2445-2451.

13.
J. Zhang and R. Lu, Strongly regular rings and SF-rings, Northeast Math. J., 14(1)(1998), 61-68.

14.
H. Zhou and X. Wang, Von Neumann regular rings and right SF-rings, Northeast Math. J., 20(2)(2004a), 75-78.

15.
H. Zhou, Left SF-rings and regular rings, Communications in Algebra, 35(2007), 3842-3850.