This paper is mainly concerned
with describing the category of all algebraically compact ( = pure-injective) modules.
A family of functors from this category to categories of injective modules, that is,
spectral categories, is defined. Via these functors we transfer the decompositions of
the objects of a spectral category and their invariants to algebraically compact
modules. For instance, as a corollary we find the decompositions and the
invariants for algebraically compact abelian groups and the decompositions for
algebraically compact modules over Prüfer rings. Our results yield a connection
between the theory of algebraically compact modules and the one of injective
modules.
