The purpose of this book is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In particular we work in the areas of chain conditions on modules, injectivity and CS (extending) modules. At the outset we acquaint the reader with the main ideas used in torsion theory. We then obtain results that fall into three more or less interrelated areas. In the first area we introduce a torsion-theoretic analogue of Max modules and generalize an important characterization of Noetherian rings by Shock. The second area deals with various flavors of relative injectivity. We generalize well-known results by Fuchs, Azumaya, Faith, Albu and Nastasescu and Cailleau. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. Our motivation is to provide a torsion-theoretic analogue of a celebrated result by Okado. This book should appeal to researchers in the theory of rings and modules with particular emphasis on torsion theory.

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