@article{Salam2023, 
author = {Saif Salam and Khaldoun Al-Zoubi},
title = {Graded modules with Noetherian graded second spectrum},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {3},
pages = {6626-6641},
keywords = {graded second submodule, graded second spectrum, graded Zariski socle, Zariski topology, Noetherian graded second spectrum},
url = {https://www.sciopen.com/article/10.3934/math.2023335},
doi = {10.3934/math.2023335},
abstract = {Let    R be a    G graded commutative ring and    M be a    G-graded    R-module. The set of all graded second submodules of    M is denoted by    S  p  e      c    G    s    (  M  )  , and it is called the graded second spectrum of    M. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate    S  p  e      c    G    s    (  M  ) with the Zariski topology from the viewpoint of being a Noetherian space.}
}