ON A GENERALIZATION OF RIGHT DUO RINGS

Title & Authors
ON A GENERALIZATION OF RIGHT DUO RINGS
Kim, Nam Kyun; Kwak, Tai Keun; Lee, Yang;

Abstract
We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right $\small{{\pi}}$-duo as a generalization of (weakly) right duo rings. Abelian $\small{{\pi}}$-regular rings are $\small{{\pi}}$-duo, which is compared with the fact that Abelian regular rings are duo. For a right $\small{{\pi}}$-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) $\small{{\pi}}$-regular ring with $J(R) Keywords right $\small{{\pi}}$-duo ring;(weakly) right duo ring;(strongly) $\small{{\pi}}$-regular ring;every prime ideal is maximal;polynomial ring;matrix 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), no. 4, 1009-1021. 3. H. H. Brungs and G. Torner, Chain rings and prime ideals, Arch. Math. (Bassel) 27 (1976), no. 3, 253-260. 4. A. M. Buhphang and M. B. Rege, Semi-commutative modules and Armendariz modules, Arab J. Math. Sci. 8 (2002), no. 1, 53-65. 5. F. Dischinger, Sur les anneaux fortement${\pi}$-reguliers, C. R. Acad. Sci. Paris, Ser. A 283 (1976), no. 8, 571-573. 6. J. L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932), no. 2, 85-88. 7. E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958), 79-91. 8. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979. 9. Y. Hirano, Some studies on strongly${\pi}$-regular rings, Math. J. Okayama Univ. 20 (1978), no. 2, 141-149. 10. Y. Hirano, C. Y. Hong, J. Y. Kim, and J. K. Park, On strongly bounded rings and duo rings, Comm. Algebra 23 (1995), no. 6, 2199-2214. 11. 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), no. 1, 35-44. 12. C. Y. Hong, N. K. Kim, and Y. Lee, Hereditary and Semiperfect Distributive Rings, Algebra Colloq. 13 (2006), no. 3, 433-440. 13. C. Y. Hong, N. K. Kim, Y. Lee, and P. P. Nielsen, Minimal prime spectrum of rings with annihilator conditions, J. Pure Appl. Algebra 213 (2009), no. 7, 1478-1488. 14. 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), no. 3, 411-422. 15. C. Huh, H. K. Kim, N. K. Kim, C. I. Lee, Y. Lee, and H. J. Sung, Insertion-of-Factors-Property and related ring properties, (submitted). 16. C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. 17. C. Huh, H. K. Kim, and Y. Lee, Examples of strongly${\pi}$-regular rings, J. Pure Appl. Algebra 189 (2004), no. 1-3, 195-210. 18. C. Huh and Y. Lee, A note on${\pi}\$-regular rings, Kyungpook Math. J. 38 (1998), no. 1, 157-161.

19.
T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.

20.
S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199.

21.
Y. C. Jeon, H. K. Kim, Y. Lee, and J. S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), no. 1, 135-146.

22.
H. K. Kim, N. K. Kim, and Y. Lee, Weakly duo rings with nil Jacobson radical, J. Korean Math. Soc. 42 (2005), no. 3, 455-468.

23.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.

24.
J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.

25.
G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123.

26.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., Chichester, New York, Brisbane, Toronto, Singapore, 1987.

27.
X. Yao, Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), no. 1-2, 19-24.

28.
H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), no. 1, 21-31.