ABSTRACT
The first section defines type submodules and describes their basic properties. In order to be able to use type submodules effectively, one has to be able to identify which submodules are type submodules, and to be familiar with a few module operations through which type submodules are obtained. Then a special kind of type submodules, those which are also atomic, are
used to define the type dimension of any module. Some of the more useful properties of the type dimension are explained. Section 4.2 develops computational formulas for calculating type dimension, and concludes with formulas for the type dimension for several classes of modules and rings, such as formal triangular matrix rings and Laurent polynomial modules and rings. Several classes of rings defined using type dimension, including those rings
with type ascending and descending chain conditions, are investigated in section 4.3 as immediate applications of the type dimension. The concept of the type dimension first appeared in [140], where many of
its properties were proved. Then it was subsequently used throughout [141]. By then, its usefulness was apparent.
