Bull. Korean Math. Soc. 2016; 53(3): 925-942
Printed May 31, 2016
https://doi.org/10.4134/BKMS.b150441
Copyright © The Korean Mathematical Society.
Nam Kyun Kim, Tai Keun Kwak, and Yang Lee
Hanbat National University, Daejin University, Pusan National University
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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