Classes of Modules

General module theory is too vast an area to provide for general structure theory, the major problem being the lack of a satisfactory decomposition theory. In this monograph, the authors propose the notions of a natural class of modules (i.e. a class closed under submodules, direct sums and essential extensions) and of a type submodule (i.e. a submodule maximal among those belonging to a natural class) to overcome this problem. They call a module M a TS-module if every type submodule of M is a direct summand. TS-modules thus generalize the notions of an extending module or a CS-module. In chapter 4, type dimension theory of a module is developed in analogy to the classical finite uniform dimension theory. Chapter 5 extends decomposition theory of CS-modules to TS-modules. It contains moreover a far reaching generalization of the Goodearl-Boyle decomposition theory of non-singular injective modules into type I, II, and III submodules (using the fact that in the appropriate generalization, type I, II, and III modules form a natural class each). Chapter 6 deals with relations between the structure of a ring R and the lattice of (pre-)natural classes of R-modules. As the authors point out, the book not only presents a new theory but it suggests a new path through general ring and module theory.

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