WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL

Title & Authors
WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL
KIM HONG KEE; KIM NAM KYUN; LEE YANG;

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 $\small{\pi}$-regular ring;quasi-duo ring; Language English Cited by 1. ON A GENERALIZATION OF RIGHT DUO RINGS,;;; 대한수학회보, 2016. vol.53. 3, pp.925-942 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 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 5. A. W. Chatters and W. Xue, On right duo p.p. rings, Glasg. Math. J. 32 (1990), 221-225 6. R. C. Courter, Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84 (1982), 157-161 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 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 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 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 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

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