New generalizations of the summand intersection property

We say that a module [Formula: see text] is an SISZ (SIPSZ) module if any nonzero intersection of two (perspective) direct summands of [Formula: see text] is not small in [Formula: see text]. We show that a finite direct sum of SISZ modules need not be an SIPSZ module. Right V-rings and right hereditary rings are characterized in terms of the properties SISZ and SIPSZ. A ring [Formula: see text] is said to be right SISZ (SIPSZ) if the right [Formula: see text]-module [Formula: see text] is SISZ (SIPSZ). We investigate products of right SISZ (SIPSZ) rings and study right SISZ (SIPSZ) trivial extension rings.