The non-modularity index quantifies the extent of interaction within a group of criteria in multiple criteria decision problems, or extent of interaction among players in a coalition in the context of cooperative games. This paper examines the properties of the non-modularity index in relation to dummy players and dummy partnerships. It establishes that the scaled non-modularity index is maximal at unanimity games and additive with respect to non-interacting (or dummy) coalitions and partnerships. This paper also provides an axiomatic characterisation of the non-modularity index based on linearity, dummy partnership additivity and maximality axioms.