On Graded S-semiprime Submodules of Graded Modules Over Graded
Commutative Rings
Mohammad Alkhatib1 , Khaldoun Al-Zoubi1,∗
1Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Box 3030,
Irbid 22110, Jordan
Emails: mwalkhatib20@sci.just.edu.jo; kfzoubi@just.edu.jo
Abstract
Let G be a group with identity e. Let T be a commutative G-graded ring with non-zero identity, W be a graded
T-module and S ⊆ h(T) a multiplicatively closed subset of T. In this article, we introduce and study the
concept of graded S-semiprime submodules. A graded submodule K of W with (K :T W) ∩ S = ∅ is said
to be graded S-semiprime, if there exists a fixed st ∈ S such that whenever rn
i mj ∈ K for some ri ∈ h(T),
mj ∈ h(W), t, i, j ∈ G, and n ∈ N, then strimj ∈ K. Some characterizations and properties of graded
S-semiprime submodules are given.
Keywords: Graded S-semiprime submodule; Graded S-semiprime ideal; Graded semiprime submodule