It was shown in the last chapter that every totally transcendental module is a direct sum of indecomposable submodules. The proof of this was short – in a sense too short, since it tells us little about the indecomposable factors which occur. For instance, if a is an element of the totally transcendental module M and if N is a minimal direct summand of M containing a, then what is the relationship between N and a? Is N uniquely determined by a? Does N depend on a or just on the pp-type of a? These questions will be answered in this chapter.

In section 1 it is shown that, given any pure-injective module M and any element (or subset) of M, there is a minimal direct summand of M containing the element (or subset). We will call this the hull, N(A), of the element or subset A and the terminology is justified by showing that this hull is unique up to isomorphism over A. Furthermore, it is shown that the hull of A depends only on the pp-type of A.

The terminology is reminiscent of that for injective hulls: indeed, the above hulls can be seen as injective hulls in an appropriate (functor) category. I don't, however, take that approach to them, preferring to work on a more “concrete” level. The injective hull of a module A is characterised by the fact that every element in it is “linked” in a non-trivial way to A by an atomic relation (equation). There is an analogy for hulls: every element of the hull of A is linked in a non-trivial way to A by a pp-relation.