On the endomorphism ring of a module with relative chain conditions

مقال فى مجلة
الغامدي, حنان العليان و احمد . 2016
نوع عمل المنشور
ورقة علمية
اسم الناشر
Communications in Algebra
مجلة/صحيفة
British Journal of Mathematics & Computer Science
تاريخ المؤتمر
ملخص المنشورات

A well-known result of Small states that if M is a noetherian left R-module having endomorphism ring S then any nil subring of S is nilpotent. Fisher [4] dualized this result and showed that if M is left artinian then any nil ideal of S is nilpotent. He gave a bound on the indices of nilpotency of nil subrings of the endomorphism rings of noetherian modules and raised the dual question of whether there are such bounds in the case of artinian modules. He gave an affirmative answer if the module is also assumed to be finitely-generated. Similar affirmative answers for modules with finite homogeneous length were given in [10] and [15]. On the other hand, the nilpotence of certain ideals of the endomorphism rings of modules noetherian relative to a torsion theory has been extensively studied. See [2,6,8,12,15,17]. Jirasko [11] dualized, in some sense, some of the results of [6] to torsion modules satisfying the descending chain conditions with respect to some radical.