A module $M$ is said to have the SIP if intersection of each pair of direct summands is also a direct summand of $M$. In this article, we define a module $M$ to have the $\operatorname{SIP}^r$ if and only if intersection of each pair of exact direct summands is also a direct summand of $M$ where $r$ is a left exact preradical for the category of right modules. We investigate structural properties of $SIP^r$-modules and locate the implications between the other summand intersection properties. We deal with decomposition theory as well as direct summands of $SIP^r$-modules. We provide examples by looking at special left exact preradicals.