Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules - Azzh Saad Alshehry; Malik Bataineh; Rashid Abu-Dawwas (Mathematics, 2021)
The main goal of this article is to explore the concepts of graded <semantics>ϕ</semantics>-2-absorbing and graded <semantics>ϕ</semantics>-2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let <semantics>ϕ:GS(M)→GS(M)⋃{∅}</semantics> be a function, where <semantics>GS(M)</semantics> denotes the collection of graded R-submodules of M. A proper <semantics>K∈GS(M)</semantics> is said to be a graded <semantics>ϕ</semantics>-2-absorbing R-submodule of M if whenever <semantics>x,y</semantics> are homogeneous elements of R and s is a homogeneous element of M with <semantics>xys∈K−ϕ(K)</semantics>, then <semantics>xs∈K</semantics> or <semantics>ys∈K</semantics> or <semantics>xy∈(K:RM)</semantics>, and we call K a graded <semantics>ϕ</semantics>-2-absorbing primary R-submodule of M if whenever <semantics>x,y</semantics> are homogeneous elements of R and s is a homogeneous element of M with <semantics>xys∈K−ϕ(K)</semantics>, then <semantics>xs</semantics> or <semantics>ys</semantics> is in the graded radical of K or <semantics>xy∈(K:RM)</semantics>. Several properties of these new forms of graded submodules are investigated.