[Paper Review] The total zero-divisor graph of commutative rings

Alen Ðurić, Sara Jevđenić+2 more|arXiv (Cornell University)|Mar 15, 2018

Rings, Modules, and Algebras7 references3 citations

TL;DR

This paper introduces the total zero-divisor graph of commutative unital rings, defined by adjacency when both the product and sum of distinct nonzero zero-divisors are zero-divisors. It characterizes Artinian rings with connected total zero-divisor graphs, computes key invariants for ℤₘ, and proves that ZT(ℤₘ) ≅ ZT(ℤₙ) if and only if m = n, establishing a complete isomorphism invariant for these rings.

ABSTRACT

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$.

Motivation & Objective

• To define and study the total zero-divisor graph, a new graph combining zero-divisor and total graph properties in commutative rings.
• To characterize Artinian rings whose total zero-divisor graphs are connected, revealing dependence on associated prime ideals and maximal ideals.
• To compute fundamental graph invariants—girth, chromatic number, domination number, metric dimension—for the total zero-divisor graph of ℤₘ.
• To establish a complete isomorphism invariant for total zero-divisor graphs of ℤₘ by proving ZT(ℤₘ) ≅ ZT(ℤₙ) if and only if m = n.
• To identify all acyclic total zero-divisor graphs and analyze vertex indistinguishability in ℤₘ.

Proposed method

• Define the total zero-divisor graph ZT(R) with vertex set Z(R)\{0}, where distinct vertices u and v are adjacent iff uv = 0 and u + v ∈ Z(R).
• Use structural ring theory, including associated primes, annihilators, and nilpotent ideals, to analyze connectivity and diameter in Artinian rings.
• Apply multiplicative number theory and divisor structure of ℤₘ to compute invariants such as girth, chromatic number, and domination number.
• Establish vertex indistinguishability classes in ZT(ℤₘ) by analyzing adjacency conditions and associativity, leading to bounds on resolving sets.
• Construct a minimal resolving set B of size m − φ(m) − τ(m) + n + 1 for ℤₘ with n prime factors, proving the metric dimension via complement argument.
• Use isomorphism invariance and divisor class analysis to show that ZT(ℤₘ) ≅ ZT(ℤₙ) implies m = n, relying on degree sequences and structural uniqueness.

Experimental results

Research questions

• RQ1 For which Artinian rings is the total zero-divisor graph connected, and how does this depend on the ring’s prime ideal structure?
• RQ2 What are the exact values of the girth, chromatic number, domination number, and metric dimension of ZT(ℤₘ) for a given m?
• RQ3 When is the total zero-divisor graph of ℤₘ acyclic, and what is the structure of such graphs?
• RQ4 What is the metric dimension of ZT(ℤₘ), and how does it depend on the prime factorization of m?
• RQ5 Under what conditions is ZT(ℤₘ) isomorphic to ZT(ℤₙ), and can this isomorphism be characterized solely by m = n?

Key findings

• The total zero-divisor graph of an Artinian ring is connected if and only if the ring has at most one associated prime ideal, and its diameter is 2 if the ring has more than one maximal ideal, otherwise 1.
• For ℤₘ with m = p₁^{m₁}⋯pₙ^{mₙ}, the girth of ZT(ℤₘ) is 3 if n ≥ 2, and 4 if n = 1.
• The chromatic number of ZT(ℤₘ) is equal to the number of distinct prime factors of m, and the domination number is τ(m) − 1, where τ(m) is the number of positive divisors of m.
• The metric dimension of ZT(ℤₘ) is m − φ(m) − τ(m) + n + 1 when m has at least two distinct prime factors, and m − φ(m) − τ(m) + 1 when m is a prime power.
• The total zero-divisor graphs ZT(ℤₘ) and ZT(ℤₙ) are isomorphic if and only if m = n, establishing a complete isomorphism invariant.
• The only indistinguishable vertices in ZT(ℤₘ) are associates and pairs of the form p_i^{m_i} and p_i^{m_i−1} for each prime power factor, which determines the minimal resolving set size.