Characterization of modules and rings with the intersection of direct summands sub-modules and the sum of their sums

The study of the relations between two direct summands was initiated by Kaplansky (1954), continues today. If the intersection of any two direct summands of an R-module M is also a direct summand, the M module is called a SIP module. SIP modules was given by Wilson (1986). In this context, the SSP module, which is considered as dual of SIP modules, was defined and introduced into the literature by Garcia (1989) as the SSP module, if the sum of two direct summands is a direct summand of M . Afterwards, many author studies were conducted regarding SSP and SIP modules. They obtained many generalizations and characterizations of the SSP and SIP module families from these studies ( Abyzov and Tuganbaev 2014 ; Alkan and Harmancı 2002; Amin, ˙Ibrahim and Yousif 2014 ; Anderson and Fuller 1974; Arnold and Hausen 1990; Clark,Lomp, Vanaja and Wisbauer 2006; Dung,Huyn,Smith and Wisbauer 1994;Fuchs 1970; Hausen 1989; Kasch 1982; Karabacak and Tercan 2007; Nicholson and Yousif 2003; Ta¸sdemir and Karabacak 2019; Valcan 2009; Wisbauer 1991). In this work, the definitions of ESIP and ESSP modules, which are generalizations of SIP and SSP modules, are given and the features of ESIP and ESSP modules are investigated. Let M be an R−module; for every A, B ≤d M if there is D ≤d M such that A ∩ B ≤e D , then M has an essential summnad intersection property and for every A, B ≤d M if there is D ≤d M such that A + B ≤e D, it is said that the M module has the essential summand sum property. Then, the differences between ESIP and ESSP modules and SIP and SSP modules are mentioned and examples are given to understand these differences. The conditions under which rings and modules would have ESIP and ESSP properties are examined separately and the conditions obtained as a result of this examination were given. Characterization of ESIP and ESSP properties on rings and modules has been investigated, and theorems regarding SIP and SSP have been generalized to ESIP and ESSP properties.