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.

[English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (565kB)

Abstract

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.

Actions (login required)

View Item