Contributions to the Theory of Continuous Modules

Abstract

<p>A module M is called continuous if (i) every submodule of M is essential in a summand of M, and (ii) if a submodule A is isomorphic to a summand of M, then A is itself a summand of M.</p> <p>Injective and quasi-injective modules play an important role in module theory and continuous modules are a generalization of these concepts. Many of the important properties that hold for (quasi-) injective modules, still hold for continuous modules, and it is often more convenient to work with the above two conditions rather than the notion of (quasi-) injectivity.</p> <p>This thesis deals with several important aspects of the theory of continuous modules. We give a decomposition theorem for continuous modules and, as a corollary, obtain a partial generalization of a result of Matlis and Papp. We also answer the open question: When is a finite direct sum of indecomposable continous modules continuous modules is also examined.</p> <p>The main chapter deals with the concept of continuous hulls. We give an appropriate definition, explicitly describe the continuous hulls for the classes of uniform cyclic modules, and of non-singular cyclic modules over commutative rings, and exhibit them by concrete examples. A necessary and sufficient condition for the existence of continuous hulls for arbitrary cyclic modules over a commutative ring is also given. In our opinion, these results constitute an important development, since these first steps towards establishing the existence of continuous hulls should stimulate further research, and since the since the knowledge of their existence should prove valuable in related investigations.</p> <p>Finally, we study in detail commutative rings for which every continuous module is quasi-injective. It is shown that this property holds true for large classes of rings such as noetherian ones, and semi-primary ones whose Jacobson radical has square zero. We characterize several other classes of rings with this property. Many examples are provided throughout the thesis which show the existence of continuous modules (and hulls) which are not (quasi-) injective.</p>