  We introduce the notion of residual intersections of modules and prove their
existence. We show that projective dimension one modules have Cohen-Macaulay
residual intersections, namely they satisfy the relevant Artin-Nagata property.
We then establish a formula for the core of orientable modules satisfying
certain homological conditions, extending previous results of Corso, Polini,
and Ulrich on the core of projective one modules. Finally, we provide examples
of classes of modules that satisfy our assumptions.
