We introduce a generalization of algebraic compactness for modules analogous to, and extending the one given for abelian groups by C. Megibben in [13]. After a preliminary section, we discuss the relationship between this notion and the matrix subgroups of W. Zimmermann [17], and then use the results obtained to generalize, through a filter-theoretic approach, several theorems of W. Zimmermann [17] and E.M. Martinez [11].