### 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 Γ_{0}(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: {R_{1}, R_{2}, …, R_{m}} and {S_{1}, S_{2}, …, 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≅R_{1} × R_{2} ×⋯ × R_{m}, S≅S_{1} × S_{2} ×⋯ × S_{t}, and neither R nor S is a finite field. We show that if Γ_{0}(R)≅ Γ_{0}(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 language | English |
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Title of host publication | Association for Women in Mathematics Series |

Publisher | Springer |

Pages | 173-179 |

Number of pages | 7 |

DOIs | |

State | Published - 2020 |

### Publication series

Name | Association for Women in Mathematics Series |
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Volume | 21 |

ISSN (Print) | 2364-5733 |

ISSN (Electronic) | 2364-5741 |

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## Cite this

*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