Graph Structures Arising from Ideals in Commutative Rings

Abstract

In the present paper, an algebraic-combinatorial approach is introduced to study ideal graphs associated with commutative rings, and standard constructions like Noetherian, Artinian, Local rings, and Principal Ideal Domains (PIDs) are specially studied. Intersection and co-intersection of proper ideals are used to build ideal graphs, and the study of graph invariant properties like chromatic number, clique number and symmetries of automorphisms can then be explored. The paper performs case studies in detail such as rings, e.g., rings with names like Z6, Z12, local Galois rings, Artinian rings with finite decompositions of the rings to see how the structure and complexity of the graph formed as a result of the algebraic details of the rings that make the structure (the number of maximal ideals, chain divisibility, and lattice morphisms). One of the main contributions of this research is the establishment and demonstration of three new theorems, namely the Ideal Projection Symmetry Theorem which entails that the automorphism group of an ideal graph splits according to the product decomposition of the ring; the Ideal Graph Entanglement Principle, which introduces a new index that measures the hierarchical complexity of ideal graphs; and the Ideal Nesting Spectrum Theorem, relating the clique number of a graph to the length of intersecting chains of ideals. this survey of ring classes shows how these invariants and theoretical constructions The resulting popsicle graph model provides new avenues of ring structure classification, which aid in enhanced research in algebraic topology, structural graph theory, and computational ring studies.

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