[Paper Review] Categorial properties of compressed zero-divisor graphs of finite commutative rings

Alen Ðurić, Sara Jevđenić+1 more|arXiv (Cornell University)|Jul 30, 2018

Rings, Modules, and Algebras24 references4 citations

TL;DR

This paper introduces a new compressed zero-divisor graph Θ(K) for finite commutative unital rings using the associatedness relation, which enables a natural functorial structure. It proves that Θ(K) preserves categorial products and fully characterizes local rings and principal ideal rings via graph isomorphisms to staircase graphs, offering a categorical framework for ring classification through graph-theoretic invariants.

ABSTRACT

We define a compressed zero-divisor graph $\varTheta(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation. We prove that this is the best possible compression which induces a functor $\varTheta$, and that this functor preserves categorial products (in both directions). We use the structure of $\varTheta(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

Motivation & Objective

• To develop a compressed zero-divisor graph Θ(K) that extends naturally to a functor on the category of finite commutative unital rings.
• To identify the coarsest equivalence relation—associatedness—that still supports such a functorial compression.
• To characterize finite commutative unital rings via the graph structure of Θ(K), particularly local rings and principal ideal rings.
• To establish that categorial products in the ring category correspond to tensor products in the graph category under Θ.
• To demonstrate that Θ(K) captures essential ring-theoretic properties, such as nilpotency index and ideal structure, through graph isomorphisms.

Proposed method

• Define Θ(K) as a graph whose vertices are associatedness classes of nonzero zero-divisors in a finite commutative unital ring K, with edges defined by zero-product relations.
• Use the associatedness relation (a ∼ b iff a = bu for unit u) to compress the zero-divisor structure, ensuring functoriality.
• Prove that Θ preserves categorial products in both directions: ring decompositions correspond to graph decompositions via tensor products.
• Introduce and analyze 'staircase graphs' SGk, which serve as building blocks for Θ(K) of local rings.
• Establish that Θ(K) is isomorphic to a finite tensor product of staircase graphs if and only if K is a principal ideal ring.
• Extend the definition of Θ(K) to infinite rings using principal ideals as vertices, with adjacency defined by zero product of ideals.

Experimental results

Research questions

• RQ1 Can the zero-divisor graph of a finite commutative unital ring be compressed via associatedness in a way that preserves categorical structure?
• RQ2 Does the functor Θ preserve categorial products in both directions, i.e., does a decomposition of Θ(K) imply a decomposition of K?
• RQ3 To what extent does the graph structure of Θ(K) determine the ring-theoretic properties of K, such as being local or a principal ideal ring?
• RQ4 Is the associatedness relation the coarsest equivalence relation that allows Θ to be a functor on finite commutative unital rings?
• RQ5 Can the nilpotency index of the maximal ideal of a finite local ring be recovered from Θ(K)?
• RQ6 Can the structure of Θ(K) distinguish between non-isomorphic finite local PIRs with isomorphic non-compressed graphs?

Key findings

• Θ(K) is the best possible compression of the zero-divisor graph via an equivalence relation that induces a functor, with associatedness being the coarsest such relation.
• The functor Θ preserves categorial products: a ring decomposition K ≅ K₁ × K₂ implies Θ(K) ≅ Θ(K₁) × Θ(K₂), and conversely, a graph decomposition implies a ring decomposition.
• A finite commutative unital ring K is local if and only if Θ(K) is isomorphic to a single staircase graph SGn for some n ≥ 0.
• A finite commutative unital ring K is a principal ideal ring if and only if Θ(K) is isomorphic to a finite tensor product of staircase graphs.
• For a finite local PIR, the index of nilpotency of its maximal ideal equals the height of the corresponding staircase graph SGn.
• There exist non-isomorphic finite local PIRs (e.g., Z₁₆, Z₂[x]/(x⁴), Z₄[x]/(x²−2), Z₄[x]/(x²−2x−2)) with isomorphic Θ(K) ≅ SG₃ and identical non-compressed zero-divisor graphs.