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Rings Characterized via a Class of Left Exact Preradicals

Part of: Radicals and radical properties of rings Rings and algebras arising under various constructions Modules, bimodules and ideals

Published online by Cambridge University Press:  17 December 2015

Noyan Er
Noyan Er*
Affiliation:
Yıldız Technical University, Department of Mathematical Engineering, Istanbul, Turkey (noyaner@yahoo.com)
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Abstract

For two modules M and N, iM (N) stands for the largest submodule of N relative to which M is injective. For any module M, iM : Mod-R → Mod-R thus defines a left exact preradical, and iM (M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, iM (R) = R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and iM is a radical for each M ∈ Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.

Keywords

Ringpreradicalinjective

MSC classification

Secondary: 16D50: Injective modules, self-injective rings 16N60: Prime and semiprime rings 16S90: Torsion theories; radicals on module categories
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 641 - 653
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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