Generalizations of graded second submodules

Let G be a group with identity e. Let R be a graded ring, I a graded ideal of R and M be a G-graded R-module. Let ψ: S^gr(M) → S ^gr(M) ∪ {∅} be a function, where S^gr(M) denote the set of all graded submodules of M. In this article, we introduce and study the concepts of graded ψ -second submodules and graded I-second submodules of a graded R-module which are generalizations of graded second submodules of M and investigate some properties of this class of graded modules.