On a generalization of right duo rings
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We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing {\it right $\pi$-duo} as a generalization of (weakly) right duo rings. Abelian $\pi$-regular rings are $\pi$-duo, which is compared with the fact that Abelian regular rings are duo. For a right $\pi$-duo ring $R$, it is shown that every prime ideal of $R$ is maximal if and only if $R$ is a (strongly) $\pi$-regular ring with $J(R)=N_*(R)$. This result may be helpful to develop several well-known results related to {\it pm} rings (i.e., rings whose prime ideals are maximal). We also extend the right $\pi$-duo property to several kinds of ring which have roles in ring theory.
Abstract
We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing {\it right $\pi$-duo} as a generalization of (weakly) right duo rings. Abelian $\pi$-regular rings are $\pi$-duo, which is compared with the fact that Abelian regular rings are duo. For a right $\pi$-duo ring $R$, it is shown that every prime ideal of $R$ is maximal if and only if $R$ is a (strongly) $\pi$-regular ring with $J(R)=N_*(R)$. This result may be helpful to develop several well-known results related to {\it pm} rings (i.e., rings whose prime ideals are maximal). We also extend the right $\pi$-duo property to several kinds of ring which have roles in ring theory.
참고문헌 (28)
- G. Azumaya, "Strongly π-regular rings", J. Fac. Sci. Hokkaido Univ, 13, pp.34~39 (1954)
- A. Badawi, "On abelian -regular rings", Comm. Algebra, 25(4), pp.1009~1021 (1997)
- H. H. Brungs, "Chain rings and prime ideals", Arch. Math. (Bassel), 27(3), pp.253~260 (1976)
- A. M. Buhphang, "Semi-commutative modules and Armendariz modules", Arab J. Math. Sci, 8(1), pp.53~65 (2002)
- F. Dischinger, "Sur les anneaux fortement π-reguliers", C. R. Acad. Sci. Paris, Ser. A, 283(8), pp.571~573 (1976)
- J. L. Dorroh, "Concerning adjunctions to algebras", Bull. Amer. Math. Soc, 38(2), pp.85~88 (1932)
- E. H. Feller, "Properties of primary noncommutative rings", Trans. Amer. Math. Soc, 89, pp.79~91 (1958)
- K. R. Goodearl, "Von Neumann Regular Rings" (1979), Pitman
- Y. Hirano, "Some studies on strongly π-regular rings", Math. J. Okayama Univ, 20(2), pp.141~149 (1978)
- Y. Hirano, "On strongly bounded rings and duo rings", Comm. Algebra, 23(6), pp.2199~2214 (1995)
- C. Y. Hong, "On weak -regularity of rings whose prime ideals are maximal", J. Pure Appl. Algebra, 146(1), pp.35~44 (2000)
- C. Y. Hong, "Hereditary and Semiperfect Distributive Rings", Algebra Colloq, 13(3), pp.433~440 (2006)
- C. Y. Hong, "Minimal prime spectrum of rings with annihilator conditions", J. Pure Appl. Algebra, 213(7), pp.1478~1488 (2009)
- 허찬, "Rings whose maximal one-sided ideals are two-sided", 대한수학회보, 39(3), pp.411~422 (2002), 대한수학회
- C. Huh, "Insertion-of-Factors-Property and related ring properties", submitted, ~
- C. Huh, "p.p. rings and generalized p.p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)
- C. Huh, "Examples of strongly π-regular rings", J. Pure Appl. Algebra, 189(1-3), pp.195~210 (2004)
- C. Huh, "A note on -regular rings", Kyungpook Math. J, 38(1), pp.157~161 (1998)
- T. W. Hungerford, "Algebra" (1974), Springer-Verlag
- S. U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302(1), pp.186~199 (2006)
- Young Cheol Jeon, "On weak Armendariz rings", 대한수학회보, 46(1), pp.135~146 (2009), 대한수학회
- 김홍기, "WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL", 대한수학회지, 42(3), pp.457~470 (2005), 대한수학회
- N. K. Kim, "Armendariz rings and reduced rings", J. Algebra, 223(2), pp.477~488 (2000)
- J. Lambek, "Lectures on Rings and Modules" (1966), Blaisdell Publishing Company
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29(5), pp.2113~2123 (2001)
- J. C. McConnell, "Noncommutative Noetherian Rings" (1987), John Wiley &Sons Ltd.
- X. Yao, "Weakly right duo rings", Pure Appl. Math. Sci, 21(1-2), pp.19~24 (1985)
- H.-P. Yu, "On quasi-duo rings", Glasgow Math. J, 37(1), pp.21~31 (1995)