Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules

The main goal of this article is to explore the concepts graded ϕ-2-absorbing and primary submodules as a new generalization 2-absorbing submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be function, where GS(M) denotes collection R-submodules M. A proper K∈GS(M) said R-submodule M if whenever x,y are homogeneous elements R s element with xys∈K−ϕ(K), then xs∈K or ys∈K xy∈(K:RM), we call K xs ys in radical xy∈(K:RM). Several properties these forms investigated.