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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.
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참고문헌 (28)
참고문헌 신청
G. Azumaya / 1954 / Strongly π-regular rings / J. Fac. Sci. Hokkaido Univ 13:34~39
A. Badawi / 1997 / On abelian -regular rings / Comm. Algebra 25(4):1009~1021
H. H. Brungs / 1976 / Chain rings and prime ideals / Arch. Math. (Bassel) 27(3):253~260
A. M. Buhphang / 2002 / Semi-commutative modules and Armendariz modules / Arab J. Math. Sci 8(1):53~65
F. Dischinger / 1976 / Sur les anneaux fortement π-reguliers / C. R. Acad. Sci. Paris, Ser. A 283(8):571~573
A. Badawi / 1997 / On abelian -regular rings / Comm. Algebra 25(4):1009~1021
H. H. Brungs / 1976 / Chain rings and prime ideals / Arch. Math. (Bassel) 27(3):253~260
A. M. Buhphang / 2002 / Semi-commutative modules and Armendariz modules / Arab J. Math. Sci 8(1):53~65
F. Dischinger / 1976 / Sur les anneaux fortement π-reguliers / C. R. Acad. Sci. Paris, Ser. A 283(8):571~573
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