TY - CHAP

T1 - A note on the uniqueness of zero-divisor graphs with loops (research)

AU - Li, Aihua

AU - Miller, Ryan

AU - Tucci, Ralph P.

N1 - Publisher Copyright:
© The Author(s) and the Association for Women in Mathematics 2020.

PY - 2020

Y1 - 2020

N2 - 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, …, Rm} and {S1, S2, …, St}, where each of the Ri or Sj 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 ×⋯ × 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 Γ ⃗ 0(R) ≅ Γ ⃗ 0(S), then R≅S.

AB - 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, …, Rm} and {S1, S2, …, St}, where each of the Ri or Sj 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 ×⋯ × 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 Γ ⃗ 0(R) ≅ Γ ⃗ 0(S), then R≅S.

UR - http://www.scopus.com/inward/record.url?scp=85089476813&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-42687-3_11

DO - 10.1007/978-3-030-42687-3_11

M3 - Chapter

AN - SCOPUS:85089476813

T3 - Association for Women in Mathematics Series

SP - 173

EP - 179

BT - Association for Women in Mathematics Series

PB - Springer

ER -