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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机译：我们研究了其主要权利理想包含某种两面理想的环的结构，引入了{ it right $ pi $ -duo}作为（弱）右二重奏环的推广。 Abelian的$ pi $-常规环是$ pi $ -duo，这与Abelian的常规环是二重奏相比。对于一个正确的$ pi $ -duo环$ R $，表明，当且仅当$ R $是一个（强烈）$ pi $规则环且具有$ J（ R）= N _ *（R）$。此结果可能有助于开发与{ it pm}环（即，其理想理想最大的环）有关的几个众所周知的结果。我们还将正确的$ pi $ -duo属性扩展到在环理论中起作用的几种环。
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