On rings with a unique proper essential right ideal

O. A. S. Karamzadeh ¹   ; M. Motamedi ¹   ; S. M. Shahrtash ¹

¹ Department of Mathematics University of Ahvaz Ahvaz, Iran

Fundamenta Mathematicae, Tome 183 (2004) no. 3, pp. 229-244

Résumé

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.

Détail

DOI : 10.4064/fm183-3-3

Keywords: right ue rings rings property title maximality right socle investigated shown semiprime ring right ue ring only regular v ring socle being maximal right ideal and only intrinsic topology non discrete hausdorff dense proper right ideals semisimple proved right self injective right ue ring local right ue ring never semiprime artin semisimple modulo its jacobson radical has unique non zero ideal observe modules krull dimension right ue rings artinian noetherian every local right ue ring contains duo subring which again local ue ring basic properties right ue rings several important examples these rings given finally observed rings semiprime right goldie rings other known rings never ue rings