Algebraically compact module

In mathematics, especially in the area of abstract algebra known as module theory and in model theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

Suppose R is a ring and M is a left R-module. Take two sets I and J, and for every i in I and j in J, an element r_ij of R such that, for every i in I, only finitely many r_ij are non-zero. Furthermore, take an element m_i of M for every i in I. These data describe a system of linear equations in M:

The goal is to decide whether this system has a solution, i.e. whether there exist elements x_j of M for every j in J such that all the equations of the system are simultaneously satisfied. (Note that we do not require that only finitely many of the x_j are non-zero here.)

Now consider such a system of linear equations, and assume that any subsystem consisting of only finitely many equations is solvable. (The solutions to the various subsystems may be different.) If every such "finitely-solvable" system is itself solvable, then we call the module M algebraically compact.

A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K is injective for every right R-module C. The module M is pure-injective if any pure injective homomorphism j : M → K splits (i.e. there exists f : K → M with fj = 1_M).

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