The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over $\mathbb{Z}$-graded principal ideal domain.

In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study the graded weakly prime ideals. We investigate the graded rings in which all graded ideals are graded weakly prime. Several properties are given, and several examples to support given propositions are constructed. We initiate the study of graded weakly total prime ideals and investigate graded rings for which every proper graded ideal is graded weakly total prime.

Let G be a group with identity e, R be a commutative G-graded ring with unity 1 and M be a G-graded R-module. In this article, we introduce and study two generalizations of graded second submodules, namely, graded 2-absorbing second submodules and graded strongly 2- absorbing second submodules. Also, we introduce and study the concept of graded quasi 2-absorbing second submodules, that is a generalization for graded strongly 2-absorbing second submodules.