On SF-rings and Regular Rings
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 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.
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 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.
참고문헌 (15)
- F. W. Anderson, "Rings and Categories of Modules" (1974), Springer -Verlag
- V. S. Ramamurthy, "Weakly regular rings", Canad. Math. Bull., 16(3), pp.317~321 (1973)
- V. S. Ramamurthy, "On the injectivity and flatness of certain cyclic modules", Proc. Amer. Math. Soc., 48, pp.21~25 (1975)
- M. B. Rege, "On von Neumann regular rings and SF-rings", Math. Japonica, 31(6), pp.927~936 (1986)
- Xianmei Song, "Generalizations of V-rings", Kyungpook Mathematical Journal, 45(3), pp.357~362 (2005), 경북대학교 자연과학대학 수학과
- B. T. Strenstrom, "Rings of quotients- An introduction to the methods of ring theory" (1975), Springer-Verlag
- G. Xiao, "On GP-V-rings and characterizations of strongly regular rings", Northeast. Math. J., 18(4), pp.291~297 (2002)
- Y. Xiao, "One sided SF-rings with certain chain conditions", Canad. Math. Bull., 37(2), pp.272~277 (1994)
- R. Yue, "On von Neumann regular rings", Proc. Edinburgh Math. Soc., 19, pp.89~91 (1974)
- R. Yue, "On von Neumann regular rings, VIII", J. Korean Math. Soc., 19(2), pp.97~104 (1983)
- J. Zhang, "Characterizations of strongly regular rings", Northeast. Math. J., 10(3), pp.359~364 (1994)
- J. Zhang, "Von Neumann regularity of SF-rings", Communications in Algebra, 21(7), pp.2445~2451 (1993)
- J. Zhang, "Strongly regular rings and SF-rings", Northeast Math. J., 14(1), pp.61~68 (1998)
- H. Zhou, "Von Neumann regular rings and right SF-rings", Northeast Math. J., 20(2), pp.75~78 (2004)
- H. Zhou, "Left SF-rings and regular rings", Communications in Algebra, 35, pp.3842~3850 (2007)