On generalizations of modules which have the SIP

An R-module M has the summand intersection property (briefly SIP) if the intersection of any two direct summands is again a direct summand. Def-inition of SIP was given by Wilson in 1986 and this definition together with its generalizations are later studied by many autors. In this study, a general-ization of SIP modules is investigated. It is defined that a module M is called i-SIP (or has i-SIP), if the intersection of every pair of direct summands of M is isomorphic to a direct summand of M. It is shown that the class of i-SIP modules is closed neither under the direct sums nor under the direct summands. It is stated some results that demonstrate under what conditions this property is inherited by direct summands and direct sums. The characterization of this property over rings and modules is investigated. Moreover, lots of useful propo-sitions obtained in SIP modules are generalized to i-SIP modules. It is not known whether a direct summand of an FI-extending module which is a generalization of CS-modules is also FI-extending. It is given an answer to the Birkenmeier, Muller and Rızvi's question (2002) that under what conditions a direct summand of an FI-extending module is an FI-extending?: And it is proved that every direct summand of an FI-extending module with i-SIP is also FI-extending. Keywords: i-SIP, SIP, SSP, Cn-module, FI-extending module