Examples
Every vector space is algebraically compact (since it is pure-injective). More generally, every injective module is algebraically compact, for the same reason.
If R is an associative algebra with 1 over some field k, then every R-module with finite k-dimension is algebraically compact. This gives rise to the intuition that algebraically compact modules are those (possibly "large") modules which share the nice properties of "small" modules.
The Prüfer groups are algebraically compact abelian groups (i.e. Z-modules).
Many algebraically compact modules can be produced using the injective cogenerator Q/Z of abelian groups. If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all group homomorphisms from H to Q/Z. This is then a left R-module, and the *-operation yields a faithful contravariant functor from right R-modules to left R-modules. Every module of the form H* is algebraically compact. 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.
Read more about this topic: Algebraically Compact Module
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