Summand intersection property on c-closed submodules

Abstract

A module C is said to have the SIP when the intersection of any pair of direct summands of C is also a summand of C. In this manuscript, we define (strongly) summand intersection property on c-closed submodules, for short (\({SSIP}^{c}\)) \({SIP}^{c}\) if and only if the intersection of any pair of c-closed direct summands of C is (fully invariant) summand of C. Also, we introduced strongly CCLS if each c-closed submodule of C is a "fully invariant summand". We illustrate the structural features of these modules and locate these implications among some of modules’ properties.