A submodule A of a left module
B (over an associative ring with 1) is pure if for any right module F, the natural
homomorphism F ⊗A → F ⊗B is injective. A module C is pure-injective if for any
module B and pure submodule A, any homomorphism from A to C extends to B.
The theory of this notion of purity and the corresponding class of pure-injectives is
developed in this paper, with special attention to modules over commutative
Noetherian rings and Prüfer rings. It is proved that pure-injective envelopes exist
and the pure-injective modules are characterized as retracts of topologically compact
modules. For this reason, the pure-injective modules are also called algebraically
compact. For modules over Prüfer rings, certain simplifications occur, due
essentially to the fact that a finitely presented module is a summand of
a direct sum of cyclic modules. Complete sets of invariants are obtained
for certain classes of algebraically compact modules over certain Prüfer
rings.
