Module version of tensorizing maps and tensor products on C^*-algebras

For C^*-algebras A, A and B where A and B are A-bimodules with compatible actions, we consider amalgamated A-module tensor product of A and B and study its relation with the C*-tensor product of A and B for the min and max norms. We introduce and study the notions of module tensorizing maps, module exactness, and module nuclear pairs of C^*-algebras in this setting. We illustrate our results for the concrete examples of C^*-algebras on inverse semigroups.