A Note on the Uniqueness of Zero-Divisor Graphs with Loops (Research)

Abstract

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, \(\overrightarrow {\Gamma }_0(R)\) denotes the directed zero-divisor graph of R that allow loops. Consider two sets of finite rings: {R1, R2, …, Rm} and {S1, S2, …, St}, where each of the Ri or Sj is either a finite field or of the form of \(\mathbb Z_{p^{\alpha }}\) with p being a prime number and α being a positive integer. Suppose that R≅R1 × R2 ×⋯ × Rm, S≅S1 × S2 ×⋯ × St, 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 \(\overrightarrow {\Gamma }_0(R)\cong \overrightarrow {\Gamma }_0(S)\), then R≅S.