{"id":43300,"date":"2026-08-31T06:17:40","date_gmt":"2026-08-31T06:17:40","guid":{"rendered":"https:\/\/uomosul.edu.iq\/en\/computerscience\/?p=43300"},"modified":"2026-08-31T06:17:40","modified_gmt":"2026-08-31T06:17:40","slug":"1-712","status":"publish","type":"post","link":"https:\/\/uomosul.edu.iq\/en\/computerscience\/2026\/08\/31\/1-712\/","title":{"rendered":"Master&#8217;s Thesis by &#8220;Noor\u00a0 Abdulmonem Ghanem&#8221; Department of Mathematics"},"content":{"rendered":"<p>Master&#8217;s Thesis by &#8220;Noor\u00a0 Abdulmonem Ghanem&#8221; Department of Mathematics<\/p>\n<p><strong>Discussion of the Master&#8217;s Thesis Submitted to the College of Computer Science and Mathematics, Department of Mathematics, Entitled:<\/strong><\/p>\n<p>\u201c<strong>The Metric Chromatic Number of Annihilating Ideal Graph of Commutative Rings<\/strong>\u201d<\/p>\n<p><strong>supervised by \u00a0Prof. Dr. Husam Qasem Mohammad <\/strong><\/p>\n<p>This thesis investigates structural and coloring\u2013theoretic properties of the annihilating\u2013ideal graph \u00a0of a commutative ring R, defined on the set of nonzero ideals of , having a nonzero annihilator, two distinct ideals being adjacent whenever their product is zero. The study focuses on two special, and widely used, families of commutative rings: local principal ideal rings (local P.I.R.) of nilpotency index , and reduced rings \u00a0expressed as a finite direct product of fields. In\u00a0 optimization lemma is proved that locates the minimum of the discontinuous function \u00a0by comparison to its continuous surrogate , and this lemma is then used, together with the known degree formula for the ideal vertices of a local ., to obtain a closed\u2013form characterization of the metric chromatic number \u00a0of the annihilating\u2013ideal graph of a local principal ideal ring. Also for a reduced ring \u00a0that is a finite product of \u00a0fields, exact formulas are derived for the degree of an arbitrary ideal vertex, the clique and chromatic numbers, and the center of , and the metric chromatic number \u00a0is computed explicitly for \u00a0with the results collected in a summary table, relating s to the number of ideal vertices, the diameter, the degree of the minimal ideals, the chromatic number, and the metric chromatic number. Taken together, these results extend the structural theory of annihilating\u2013ideal graphs and provide new coloring invariants for two important classes of commutative rings<\/p>\n<p>The discussion committee consists of<\/p>\n<ul>\n<li>Raeda Dawood Mahmoud <em>(Chairman)<\/em><\/li>\n<li>Ahmed Amer Mohammed <em>(Member)<\/em><\/li>\n<li>Shaimaa Hatem Mohammed <em>(Member)<\/em><\/li>\n<li>Husam Qasem Mohammad \u00a0<em>(Member and Supervisor)<\/em><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Master&#8217;s Thesis by &#8220;Noor\u00a0 Abdulmonem Ghanem&#8221; Department of Mathematics Discussion of the Master&#8217;s Thesis Submitted to the College of Computer Science and Mathematics, Department of Mathematics, Entitled: \u201cThe Metric Chromatic Number of Annihilating Ideal Graph of Commutative Rings\u201d supervised by \u00a0Prof. Dr. Husam Qasem Mohammad This thesis investigates structural and coloring\u2013theoretic properties of the annihilating\u2013ideal graph \u00a0of a commutative ring R, defined on the set of nonzero ideals of , having a nonzero annihilator, two distinct ideals being adjacent whenever their product is zero. The study focuses on two special, and widely used, families of commutative rings: local principal ideal rings <a href=\"https:\/\/uomosul.edu.iq\/en\/computerscience\/2026\/08\/31\/1-712\/\"> [Read More]<\/a><\/p>\n","protected":false},"author":24,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-43300","post","type-post","status-publish","format-standard","hentry","category-viva"],"_links":{"self":[{"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/posts\/43300","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/users\/24"}],"replies":[{"embeddable":true,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/comments?post=43300"}],"version-history":[{"count":1,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/posts\/43300\/revisions"}],"predecessor-version":[{"id":43302,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/posts\/43300\/revisions\/43302"}],"wp:attachment":[{"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/media?parent=43300"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/categories?post=43300"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/uomosul.edu.iq\/en\/computerscience\/wp-json\/wp\/v2\/tags?post=43300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}