On rings with a unique proper essential right ideal
Right ue-rings (rings with the property of the title, i.e., with the maximality of the right socle) are investigated. It is shown that a semiprime ring $R$ is a right ue-ring if and only if $R$ is a regular V-ring with the socle being a maximal right ideal, and if and only if the intrinsic topology of $R$ is non-discrete Hausdorff and dense proper right ideals are semisimple. It is proved that if $R$ is a right self-injective right ue-ring (local right ue-ring), then $R$ is never semiprime and is Artin semisimple modulo its Jacobson radical ($R$ has a unique non-zero left ideal). We observe that modules with Krull dimension over right ue-rings are both Artinian and Noetherian. Every local right ue-ring contains a duo subring which is again a local ue-ring. Some basic properties of right ue-rings and several important examples of these rings are given. Finally, it is observed that rings such as $C(X)$, semiprime right Goldie rings, and some other well known rings are never ue-rings.