May 16-22, 2005

These lectures will develop the fundamentals of type submodules by two independent or parallel descriptions. One of these uses natural classes of R-modules, which are of independent interest.
For a fixed ring R, a class of modules is a natural class, or a type, if it is closed under isomorphic copies, (i) submodules, (ii) arbitrary direct sums, and (iii) injective hulls.  The class of all natural classes forms a complete Boolean lattice N(R). This lattice defines various intrinsic new module classes. For example, an R-module A is atomic if the natural class it generates is an atom in the lattice N(R). Or, a module is bottomless if it does not contain any atomic modules. These modules, the atomic ones, generalize the uniform modules, and one of their applications is to define a dimension similar to the finite Goldie dimension based on uniform modules. This opens up the study of rings and modules satisfying finiteness conditions based on this new dimension, but not satisfying ordinary finiteness conditions: ascending or descending chain conditions, or finite Goldie dimension.  Also, N(R) can be made into a functor N( -).
Historically, first module structure and decomposition theorems were proved mainly for injective nonsingular modules. Later, similar theorems were proved under restrictive hypotheses only on the complement submodules (e.g. extending modules). Just as in the case of finiteness conditions, recently it also has been discovered that it is actually enough to put restrictive hypotheses only on very special kinds of complement submodules, namely the so called type submodules. The applications and usefulness of type submodules goes far beyond module decompositions.
Emphasis will be on a logically coherent development of the fundamentals, and these topics should be easily accessible.