On a generalization of right duo rings
Bull. Korean Math. Soc. 2016 Vol. 53, No. 3, 925-942
https://doi.org/10.4134/BKMS.b150441
Published online May 31, 2016
Nam Kyun Kim, Tai Keun Kwak, and Yang Lee
Hanbat National University, Daejin University, Pusan National University
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.
Keywords : right $\pi$-duo ring, (weakly) right duo ring, (strongly) $\pi$-regular ring, every prime ideal is maximal, polynomial ring, matrix ring
MSC numbers : Primary 16D25, 16N20; Secondary 16N40, 16S36
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