Recently, Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski's Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider setting than the category of topological spaces. It is the purpose of this work to set up and apply this categorical interpretation of compactness in categories of R—modules. We obtain a theory of compactness for each torsion theory T, and in the case that the torsion theory T is hereditary, and R is, for example, noetherian and left hereditary, obtain a characterization of T-compact modules: a module G is T-compact provided G/TG is a T-injective module. Furthermore, under relatively mild assumptions on the ring R, the class of T—compact modules forms a torsion class for a torsion theory which we identify in the lattice of all torsion theories.
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