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WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL
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 Title & Authors
WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL
KIM HONG KEE; KIM NAM KYUN; LEE YANG;
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 Abstract
Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;
 Keywords
weakly duo ring;duo ring;Artinian ring;Jacobson radical;strongly -regular ring;quasi-duo ring;
 Language
English
 Cited by
 References
1.
G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. 13 (1954), 34-39

2.
A. Badawi, On abelian $\pi$-regular rings, Comm. Algebra 25 (1997), 1009-1021 crossref(new window)

3.
H. Bass, Finitistic homological dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488

4.
G. F. Birkenmeier, J. Y. Kim, and J. K. Park, A connection between weak regularity and the simplicity of prime factor rings, Proc. Amer. Math. Soc. 122 (1994), 53-58 crossref(new window)

5.
A. W. Chatters and W. Xue, On right duo p.p. rings, Glasg. Math. J. 32 (1990), 221-225 crossref(new window)

6.
R. C. Courter, Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84 (1982), 157-161 crossref(new window)

7.
F. Dischinger, Sur les anneaux fortement $\pi$-reguliers, C. R. Math. Acad. Sci. Paris 283 (1976), 571-573

8.
K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979

9.
K. R. Goodearl and R. B. Warfield. JR., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989

10.
C. Y. Hong, N. K. Kim, T. K. Kwak, and Y. Lee, On weak $\pi$-regularity of rings whose prime ideals are maximal, J. Pure Appl. Algebra 146 (2000), 35-44 crossref(new window)

11.
C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), 37-52 crossref(new window)

12.
C. Huh, S. H. Jang, C. O. Kim, and Y. Lee, Rings whose maximal one-sided ideals are two-sided, Bull. Korean Math. Soc. 39 (2002), 411-422

13.
C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751-761 crossref(new window)

14.
N. Jacobson, The Theory of Rings, American Mathematical Society, 1943

15.
J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, 1966

16.
Y. Lee and C. Huh, On rings in which every maximal one-sided ideal contains a maximal ideal, Comm. Algebra 27 (1999), 3969-3978 crossref(new window)

17.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., 1987

18.
L. H. Rowen, Ring Theory, Academic Press, Inc., 1991

19.
R. M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 36 (1948), 173-183

20.
H. Tominaga, Some remarks on$\pi$-regular rings of bounded index, Math. J. Okayama Univ. 4 (1955), 135-141 crossref(new window)

21.
X. Yao, Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), 19-24

22.
H. -P. Yu, On quasi-duo rings, Glasg. Math. J. 37 (1995), 21-31 crossref(new window)