Abstract
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 SIPr 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 SIPr-modules and locate the implications between the other summand intersection properties. We deal with decomposition theory as well as direct summands of SIPr-modules. We provide examples by looking at special left exact preradicals.
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