Master’s Thesis by “Noor  Abdulmonem Ghanem” Department of Mathematics

Discussion of the Master’s Thesis Submitted to the College of Computer Science and Mathematics, Department of Mathematics, Entitled:

“The Metric Chromatic Number of Annihilating Ideal Graph of Commutative Rings”

supervised by  Prof. Dr. Husam Qasem Mohammad

This thesis investigates structural and coloring–theoretic properties of the annihilating–ideal graph  of 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  expressed as a finite direct product of fields. In  optimization lemma is proved that locates the minimum of the discontinuous function  by 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–form characterization of the metric chromatic number  of the annihilating–ideal graph of a local principal ideal ring. Also for a reduced ring  that is a finite product of  fields, 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  is computed explicitly for  with 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–ideal graphs and provide new coloring invariants for two important classes of commutative rings

The discussion committee consists of

• Raeda Dawood Mahmoud (Chairman)
• Ahmed Amer Mohammed (Member)
• Shaimaa Hatem Mohammed (Member)
• Husam Qasem Mohammad  (Member and Supervisor)