We say that an R-module (abelian group) M has the direct summand in- tersection property (in short D.S.I.P.) if the intersection of any two direct sum- mands of M is again a direct summand in M. In this work we will present three classes of abelian groups (torsion, divisible, respectively torsion-free) which have the property that any proper subgroup has D.S.I.P. and we are going to show that there are not such mixed groups.