Algebraically Compact Module - Definitions

Definitions

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 rij of R such that, for every i in I, only finitely many rij are non-zero. Furthermore, take an element mi of M for every i in I. These data describe a system of linear equations in M:

for every i∈I.

The goal is to decide whether this system has a solution, i.e. whether there exist elements xj 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 xj 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 = 1M).

It turns out that a module is algebraically compact if and only if it is pure-injective.

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