A Poset-Theoretic Approach to Reduced Cozero-Divisor Graphs of ℤ n and Related Rings

Abstract

Let R be a finite commutative ring with identity, and let Γ′ᵣ(R) denote its reduced cozero-divisor graph. We organize Γ′ᵣ(R) through the poset Prin*(R) of nonzero proper principal ideals and prove that adjacency is exactly incomparability in this poset. Consequently, cliques correspond to antichains, independent sets correspond to chains, and the chromatic number is determined by the width through Dilworth's theorem, in particular, every such graph is perfect. For the cyclic ring Z n , the prime-power factorization of n identifies the full ideal lattice with a finite product of chains. A symmetric-chain decomposition of this product then shows that its width is the largest rank size, equivalently the central coefficient of its rank-generating polynomial. This yields explicit formulas for the number of vertices, clique number, chromatic number, and independence number of Γ′ᵣ( Z n ), together with specialized descriptions for prime, prime-power, and squarefree moduli. We also derive corresponding results for rings whose principal ideals form a chain and for finite Boolean rings. Finally, examples demonstrate both the computational strength and the reconstruction limits of the poset model: distinct rings may have isomorphic reduced graphs, whereas other small rings remain distinguishable. These results provide a unified order-theoretic framework for reduced cozero-divisor graphs and reduce several graph-invariant problems to finite-poset calculations.