In this paper, we define a module M . to be SIPz if and only if intersection of each pair of z-closed direct summands is also a direct summand of M. We investigate structural properties of SIPz-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of SIPz -modules. We apply our results to matrix rings. To this end, it is obtained that the SIPz property is not Morita invariant..