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

Aihua Li, Ryan Miller, Ralph P. Tucci

Research output: Chapter in Book/Report/Conference proceedingChapter

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, 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.

Original languageEnglish
Title of host publicationAssociation for Women in Mathematics Series
PublisherSpringer
Pages173-179
Number of pages7
DOIs
StatePublished - 2020

Publication series

NameAssociation for Women in Mathematics Series
Volume21
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741

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    Li, A., Miller, R., & Tucci, R. P. (2020). A note on the uniqueness of zero-divisor graphs with loops (research). In Association for Women in Mathematics Series (pp. 173-179). (Association for Women in Mathematics Series; Vol. 21). Springer. https://doi.org/10.1007/978-3-030-42687-3_11