Bulletin of the
Korean Mathematical Society BKMS

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