@article{Alnimer2024, 
author = {Malak Alnimer and Khaldoun Al-Zoubi and Mohammed Al-Dolat},
title = {On graded weakly        J          g      r      -semiprime submodules},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {12315-12322},
keywords = {graded weakly Jgr-semiprime submodule, graded Jgr-semiprime submodule, graded weakly semiprime submodule},
url = {https://www.sciopen.com/article/10.3934/math.2024602},
doi = {10.3934/math.2024602},
abstract = {Let    Γ be a group,        A   be a    Γ-graded commutative ring with unity    1  , and        D   a graded        A  -module. In this paper, we introduce the concept of graded weakly        J          g      r      -semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly        J          g      r       -semiprime submodules. For example, we give a characterization of graded weakly        J          g      r      -semiprime submodules. Also, we find some relations between graded weakly        J          g      r      -semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly        J          g      r      -semiprime submodules are investigated. A proper graded submodule    U of        D   is said to be a graded weakly        J          g      r      -semiprime submodule of        D   if whenever        r          g        ∈  h  (      A    )  ,        m          h        ∈  h  (      D    ) and    n  ∈            Z              +       with    0  ≠      r          g              n            m          h        ∈  U, then        r          g            m          h        ∈  U  +      J          g      r        (      D    ), where        J          g      r        (      D    ) is the graded Jacobson radical of        D    .}
}