The solutions to these systems allow the extension of certain kinds of module homomorphisms.
All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.
The module M is algebraically compact if, for all such systems, if every subsystem formed by a finite number of the equations has a solution, then the whole system has a solution.
More generally, every injective module is algebraically compact, for the same reason.
The rational numbers are algebraically compact as a Z-module.
Many algebraically compact modules can be produced using the injective cogenerator Q/Z of abelian groups.
Furthermore, there are pure injective homomorphisms H → H**, natural in H. One can often simplify a problem by first applying the *-functor, since algebraically compact modules are easier to deal with.
Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G. Every R-module is elementary equivalent to an algebraically compact R-module and to a direct sum of indecomposable algebraically compact R-modules.