Department of Mathematics
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A module is super-decomposable if it has no direct sum indecomposable direct summands.

I will start by working through some parts of two basic examples and I will try to explain why the existence/non-existence of superdecomposable pure-injective modules is an important problem in module theory (basically, if a ring has superdecomposable pure-injectives, its category of modules has to be extremely complicated).

It follows from complexity arguments in the model theory of modules that even some quite innocent looking rings (such as (Z/8)[x:x²=0], which has only 64 elements) have superdecomposables.

The model theoretic argument is rather mysterious in some ways. Two measures of complexity on the lattice of positive primitive formulas are considered: "width" and "breadth". If R has a superdecomposable pure-injective, then the lattice of pp formulas does not have "width" ["breadth"]. If R is countable, then the converse holds. It is this latter part which is quite odd, since although the countability assumption is essential to the proof as it stands, it doesn't seem to have anything essential to do with the conclusion.

I will discuss these matters quite informally.