Authors’ summary: 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.