On rings with a unique proper essential right ideal

O. A. S. Karamzadeh, M. Motamedi, and S. M. Shahrtash. "On rings with a unique proper essential right ideal." Fundamenta Mathematicae 183.3 (2004): 229-244. <http://eudml.org/doc/283387>.

@article{O2004,
abstract = {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.},
author = {O. A. S. Karamzadeh, M. Motamedi, S. M. Shahrtash},
journal = {Fundamenta Mathematicae},
keywords = {right ue-rings; uem-rings; intrinsic topologies; duo rings; isolated maximal right ideals; right socles; local rings},
language = {eng},
number = {3},
pages = {229-244},
title = {On rings with a unique proper essential right ideal},
url = {http://eudml.org/doc/283387},
volume = {183},
year = {2004},
}

TY - JOUR
AU - O. A. S. Karamzadeh
AU - M. Motamedi
AU - S. M. Shahrtash
TI - On rings with a unique proper essential right ideal
JO - Fundamenta Mathematicae
PY - 2004
VL - 183
IS - 3
SP - 229
EP - 244
AB - 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.
LA - eng
KW - right ue-rings; uem-rings; intrinsic topologies; duo rings; isolated maximal right ideals; right socles; local rings
UR - http://eudml.org/doc/283387
ER -