F-CS-Rickart Modules

Abstract

Let M be a module over a ring with identity and F be a fully invariant submodule of M. A module M is an F -CS-Rickart module if ⁻¹(F) is an essential submodule of a direct Summand of M for any  ∈ End (M). In addition, M is an F - dual – CS – Rickart module if  (F) lies above in a direct summand of M for any  ∈ End (M). In this dissertation, some properties and characterizations of F – CS–Rickart modules and F –dual-CS– Rickart modules are investigated. Moreover, we prove that any F –dual-CS– Rickart modules and F -dual-CS-Rickart modules can be written as a direct sum of two submodules such that one of them relates to F and the other one is a CS-Rickart module (dual-CS-Ruckart module). Furthermore. we study F-CS– Rickart modules when they are projective modules. In particular, we explore Z (M)-CS-Rickart modules, Z₂ (M)-CS-Rickart modules and Z* (M)-CS-Rickart modules.