Recently, Herrlich, Salicrup, and Strecker HSS have shown that Kuratowski#x2019;s Theorem, namely, that a space X is compact if and only if for every space Y, the projection #x3C0;2X#xD7;Y #x2192; 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#x2014;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#x2014;compact modules forms a torsion class for a torsion theory which we identify in the lattice of all torsion theories.
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