Graded <i>S</i> -Semiprime Ideals and Graded Weakly <i>S</i> -Semipime Ideals of Graded Rings

Let R be a graded ring and S be a multiplicative closed subset of h(R). In this paper, we introduce the concepts of graded S-semiprime ideals and graded weakly S-semiprime ideals of R. A graded ideal I of R disjoint with S is called a graded Ssemiprime ideal (respectively, graded weakly S-semiprime ideal) of R if there exists an s ∈ S such that for all x ∈ h(R), if x 2 ∈ I (respectively, 0 ≠ x 2 ∈ I), then sx ∈ I. We show that graded S-semiprime ideals and graded weakly S-semiprime ideals enjoy analogs of many fundamental properties of graded semiprime ideals and we study their characterizations in the graded ring R.