A note on the uniqueness of zero-divisor graphs with loops (research)

It is known that rings which have isomorphic zero-divisor graphs are not necessarily isomorphic. Zero-divisor graphs for rings were originally defined without loops because edges are only defined on pairs of distinct nonzero zero-divisors. In this paper, we study zero-divisor graphs of a ring R that may have loops. We denote such graphs by Γ₀(R). If R is a noncommutative ring, Γ ⃗ ₀(R) denotes the directed zero-divisor graph of R that allow loops. Consider two sets of finite rings: {R₁, R₂, …, R_m} and {S₁, S₂, …, S_t}, where each of the R_i or S_j is either a finite field or of the form of ℤpα with p being a prime number and α being a positive integer. Suppose that R≅R₁ × R₂ ×⋯ × R_m, S≅S₁ × S₂ ×⋯ × S_t, and neither R nor S is a finite field. We show that if Γ₀(R)≅ Γ₀(S), then R≅S. We further investigate directed zero-divisor graphs with loops of upper triangular matrices over finite fields. We claim that if R and S are two n by n upper triangular matrices over finite fields such that Γ ⃗ ₀(R) ≅ Γ ⃗ ₀(S), then R≅S.