<?xml version="1.0" encoding="UTF-8"?>
<article article-type="Research Article">
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher">journal-of-discrete-mathematical-sciences-and-cryptography</journal-id>
      <journal-title-group>
        <journal-title>Journal of Discrete Mathematical Sciences and Cryptography</journal-title>
      </journal-title-group>
      <issn publication-format="electronic">2169-0065</issn>
      <issn publication-format="print">0972-0529</issn>
      <publisher>
        <publisher-name>Taru Publications</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.47974/JDMSC-2458</article-id>
      <title-group>
        <article-title>Graded Le-Jgr-prime submodules in commutative graded rings</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <name>
            <surname>Alghueiri</surname>
            <given-names>Shatha</given-names>
          </name>
          <aff>Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid, P. O. Box 3030, 22110, Jordan</aff>
        </contrib>
        <contrib contrib-type="author" corresp="yes">
          <name>
            <surname>Al-Zoubi</surname>
            <given-names>Khaldoun</given-names>
          </name>
          <aff>Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid, P. O. Box 3030, 22110, Jordan</aff>
        </contrib>
      </contrib-group>
      <volume>29</volume>
      <issue>3</issue>
      <fpage>1297</fpage>
      <lpage>1305</lpage>
      <pub-date date-type="pub">
        <day>13</day>
        <month>02</month>
        <year>2026</year>
      </pub-date>
      <abstract>
        <p>In this paper, we introduce the concept of gr-Le-Jgr-PSub in the context of graded commutative rings and graded modules. These submodules extend and unify the properties of gr-Le-PSub and gr-J_gr-PSub through a fixed gr-Id L=⊕g∈Ω Lg. The research develops foundational definitions, proves equivalences, and examines the structural implications of these submodules under various module-theoretic operations, including localization, direct products, and quotient modules. Examples are provided to illustrate the distinct nature of graded Le-Jgr-prime submodules compared to other graded submodules. Furthermore, conditions under which these submodules exhibit maximal and semisimple properties are characterized, contributing new insights to the theory of graded modules. </p>
      </abstract>
      <kwd-group>
        <kwd>Graded Le-Jgr-prime submodules</kwd>
        <kwd>Graded Jgr-prime submodules</kwd>
        <kwd>Graded prime submodules</kwd>
      </kwd-group>
      <custom-meta-group>
        <custom-meta>
          <meta-name>access</meta-name>
          <meta-value>open</meta-value>
        </custom-meta>
        <custom-meta>
          <meta-name>retracted</meta-name>
          <meta-value>no</meta-value>
        </custom-meta>
      </custom-meta-group>
    </article-meta>
  </front>
</article>
