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

Abstract

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