Abstract
The main goal of this article is to explore the concepts of graded phi-2-absorbing and graded phi-2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let phi:GS(M)-> GS(M)boolean OR{empty set} be a function, where GS(M) denotes the collection of graded R-submodules of M. A proper K is an element of GS(M) is said to be a graded phi-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 is an element of K-phi(K), then xs is an element of K or ys is an element of K or xy is an element of(K:M-R), and we call K a graded phi-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 is an element of K-phi(K), then xs or ys is in the graded radical of K or xy is an element of(K:M-R). Several properties of these new forms of graded submodules are investigated.