Essentially quasi-duo rings

A ring with identity is called essentially right quasi-duo if every essential maximal right ideal of it is a two-sided ideal. Essentially right quasi-duo rings generalize essentially right duo rings, a notion that arose in the study of hypercyclic rings, and right quasi-duo rings, as introduced by S.H. Brown. We prove that a ring R R is essentially right quasi-duo if and only if R R is semisimple or R / S o c ( R R ) R/Soc(R_R) is right quasi-duo. Although it is still unknown, whether a right quasi-duo ring is left quasi-duo, we provide an example of an essentially right quasi-duo ring that is not essentially left quasi-duo. Furthermore, while exchange right quasi-duo rings are known to be clean, there exist exchange essentially right quasi-duo rings that are not clean. A thorough study of essentially right quasi-duo rings is carried out and their relationship to skew power series rings, trivial extensions and formal triangular matrix rings is explored.