# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## On rings with a unique proper essential right ideal

### Tom 183 / 2004

Fundamenta Mathematicae 183 (2004), 229-244 MSC: Primary 16L30, 16L50; Secondary 16P20, 16N60. DOI: 10.4064/fm183-3-3

#### Streszczenie

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.

#### Autorzy

• O. A. S. KaramzadehDepartment of Mathematics
University of Ahvaz
Ahvaz, Iran
e-mail
• M. MotamediDepartment of Mathematics
University of Ahvaz
Ahvaz, Iran
• S. M. ShahrtashDepartment of Mathematics
University of Ahvaz
Ahvaz, Iran

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