In this paper, we introduce and study graded 2-absorbing and graded weakly 2-absorbing ideals of a graded ring which are different from 2-absorbing and weakly 2-absorbing ideals. We give some properties and characterizations of these ideals and their homogeneous components. We investigate graded (weakly) 2-absorbing ideals of R1 × R2 where R1 and R2 are two graded rings.

<abstract> <p>In this article, we introduce the concepts of graded $ s $-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $ s $-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of a grading by a finite group, polycyclic-by-finite group, or by $ \mathbb{Z} $, in addition to the interesting case of crossed product grading, which includes the class of group rings.</p> </abstract>

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.

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