The goal of this article is to present the graded weakly S-primary ideals and g-weakly S-primary ideals which are extensions of graded weakly primary ideals. Let R be a commutative graded ring, S ⊆ h(R) and P be a graded ideal of R. We state P is a graded weakly S-primary ideal of R if there exists s ∈ S such that for all x, y ∈ h(R), if 0 = xy ∈ P , then sx ∈ P or sy ∈ Grad(P ) (the graded radical of P ). Several properties and characteristics of graded weakly S-primary ideals as well as graded g-weakly S-primary ideals are investigated.Grad(P ) = x = g∈G x g ∈ R : for all g ∈ G, there exists n g ∈ N such that x g ng ∈ P .Note that Grad(P ) is always a graded ideal of R (see [10]).Let P be a graded ideal of R and P = R, then P is said to be graded primary ideal of R, if x, y ∈ h(R) with xy ∈ P then x ∈ P or y ∈ Grad(P ) (see [12]). The concept of a graded primary ideal can be generalized in a many ways, for example, Bataineh in [5] and Atani in [3]. A proper graded ideal P of R is claimed to be a graded weakly primary ideal if whenever 0 = xy ∈ P for some x, y ∈ h(R), then either x ∈ P or y ∈ Grad(P ).

The goal of this article is to present the graded weakly S-primary ideals and graded g-weakly S-primary ideals which are extensions of graded weakly primary ideals. We state P is a graded weakly S-primary ideal of R if there exists s ∈ S such that for all x, y ∈ h(R), if 0 6=xy ∈ P, then sx ∈ P or sy ∈ Grad(P). Several properties and characteristics of graded weakly S-primary ideals as well as graded g-weakly S-primary ideals are investigated.

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The purpose of this article is to introduce the graded classical S-primary submodules which are extensions of graded classical primary submodules. We state that P is a graded classical S-primary submodule of R-module M if there exists s ∈ S such that x, y ∈ h(R) and m ∈ h(M ), if xym ∈ P , then sxm ∈ P or sy n m ∈ P for some positive integer n. Several properties and characteristics of graded classical S-primary submodules have been studied.