Rings whose cyclics satisfy a certain property

Nguyen, Hau Xuan (2018) Rings whose cyclics satisfy a certain property. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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In 1964, Osofsky proved that a ring R is semisimple artinian if and only if every cyclic right R-module is injective. Motivated by this result, there have been numerous studies in the rings whose cyclic modules satisfy a certain generalized injectivity condition. An up-to-date account of the literature on this subject can be found in [26]. Following this direction, in Chapter 2, we study the rings whose cyclics are C3-modules (or CC3-rings). We prove that a ring R is semisimple artinian if every 3-generated right R-module is a C3-module. Structure theorems of semiperfect CC3-rings and self-injective regular CC3-rings are obtained. Applications to rings whose 2-generated modules are C3-modules, and whose cyclics are quasi-continuous, are also addressed. In Chapter 3, we present basic properties of CD3-rings, i.e., rings whose cyclics are D3-modules. We show that a ring R is semisimple artinian if every 2-generated right R-module is a D3-module. Structure of self-injective regular CD3-rings is given. We characterize the rings whose cyclic modules are quasi-discrete and, respectively, discrete. In Chapter 4, we prove that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. This result is then used to characterize rings whose cyclics are lifting. New characterizations of artinian serial rings with Jacobson radical square-zero are obtained. Furthermore, we show that every cyclic right R-module is �-supplemented if and only if every cyclic right R-module is a direct sum of local modules, and that artinian serial rings are exactly these rings for which every left and right module is a direct sum of local modules. In the last chapter, we present various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our method can be adapted to generalize several known results of Mares and Nicholson from projective semiperfect modules to arbitrary semiperfect modules.

Item Type: Thesis (Doctoral (PhD))
URI: http://research.library.mun.ca/id/eprint/13472
Item ID: 13472
Additional Information: Includes bibliographical references (pages 115-119).
Keywords: Cyclic Module, C3-module, D3-module, Semiperfect Ring, Semiperfect Module, Self-injective Regular Ring, Projective Cover Preserving Direct Summands, Artinian Serial Ring, Delta-Lifting Module, Delta-Semiperfect Module
Department(s): Science, Faculty of > Mathematics and Statistics
Date: August 2018
Date Type: Submission
Library of Congress Subject Heading: Artin rings; Modules (Algebra).

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