Unit groups and graph invariants of strongly unital finite rings

Abstract

This paper provides a comprehensive algebraic and graph-theoretic analysis of finite commutative strongly unital rings, where every proper subring has a multiplicative identity distinct from that of the whole ring. We prove that such a ring is necessarily of the form R ≅ ∏ᵢ₌₁ᵏ ℤₚᵢ, with k ≥ 2 and distinct primes p₁, p₂, ..., pₖ, and that all subrings are precisely the products Sⱼ = ∏ᵢ∈ᴵⱼ ℤₚᵢ × ∏ᵢ∉ᴵⱼ {0}, indexed by subsets I ⊆ {1, ..., k}. For each such subring, we explicitly determine the structure of its unit group as Sᵢ× = ∏ᵢ∈ᴵ ℤₚᵢ×, yielding cyclic factors of orders pᵢ − 1, and provide exact formulas for the order and cyclicity criteria. We then give a complete characterization of the zero-divisor graph Γ(Sᵢ), showing that adjacency is governed by the disjointness of supports; in particular, for k = 2 the graph is complete bipartite, while for general k it has clique and chromatic numbers equal to k, diameter 2 for k ≤ 3 and diameter 3 for k ≥ 4, and binding number 1/∏ᵢ₌₁ᵏ (pᵢ − 1). Extending beyond structural descriptions, we compute a broad range of advanced graph invariants for the zero-divisor graphs of these rings, including metric dimension (h − 1), domination number (h), Roman domination number (h), matching number (⌊|V|/2⌋), multiset dimension (1 for h = 2, 2 for h ≥ 3), Wiener index, and the first and second Zagreb indices, with explicit closed-form formulas in terms of the prime factors. The fractional metric dimension and independent domination number are identified as open problems for h ≥ 3. We further demonstrate that strongly unitality forces commutativity and excludes infinite integral domains, matrix rings Mₙ(R) for n ≥ 2, and group rings R[G] for nontrivial finite groups G. Finally, we provide efficient algorithms for testing strong unitality of a finite commutative ring and for enumerating all its subrings in O(2ᵏ · k) time. The results establish a complete dictionary between the algebraic structure of strongly unital rings and the combinatorial properties of their associated graphs, revealing deep interconnections between ring theory, group theory, and graph theory.