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Abstract

Let $\mathfrakB$ be a graded commutative ring with unity, and let $\mathfrakW$ be a graded unital $\mathfrakB$-module. This study introduces and develops the concept of graded strongly $J_gr^Soc$-2-absorbing submodules, a natural extension of graded $J_gr$-2-absorbing submodules within the framework of graded module theory. The motivation for this generalization stems from the need to better capture the interplay between graded algebraic structures and the behaviors of certain radicals and socles under graded operations. A properly graded submodule $N$ of $\mathfrakW$ is defined as a graded strongly $J_gr^Soc$-2-absorbing submodule if, for all $b,uin h( \mathfrakB )$ and $cin h( \mathfrakW )$, the containment $bucin N$ implies that at least one of the following conditions holds: $bcin N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) )$, $ucin N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) )$, or $buin ( N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) ):_\mathfrakB\mathfrakW )$. Several fundamental properties of these submodules are established, along with characterizations that distinguish them from related graded structures. Moreover, the investigation reveals meaningful connections between these submodules and the graded socle and graded Jacobson radical of the module, offering new insights into their algebraic significance.

Keywords

Graded strongly $ J_ gr^ Soc$-2-absorbing submodule, Graded $ J_ gr$-2-absorbing submodule, Graded $ J^ gr$-2-absorbing submodule, Graded 2-absorbing submodule, Graded prime submodules

Subject Area

Mathematics

Article Type

Article

First Page

1020

Last Page

1026

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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