### Abstract

Given a commutative ring R, one can associate with R an undirected graph Γ(R) whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are joined by an edge iff xy = 0. In this article, we determine precisely those planar graphs that can be realized as Γ(R) when R is an infinite commutative ring.

Original language | English (US) |
---|---|

Pages (from-to) | 171-180 |

Number of pages | 10 |

Journal | Communications in Algebra |

Volume | 35 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2007 |

### Fingerprint

### Keywords

- Zero-divisor graphs

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*35*(1), 171-180. https://doi.org/10.1080/00927870601041458

**Infinite planar zero-divisor graphs.** / Smith, Neal O.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 35, no. 1, pp. 171-180. https://doi.org/10.1080/00927870601041458

}

TY - JOUR

T1 - Infinite planar zero-divisor graphs

AU - Smith, Neal O

PY - 2007/1/1

Y1 - 2007/1/1

N2 - Given a commutative ring R, one can associate with R an undirected graph Γ(R) whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are joined by an edge iff xy = 0. In this article, we determine precisely those planar graphs that can be realized as Γ(R) when R is an infinite commutative ring.

AB - Given a commutative ring R, one can associate with R an undirected graph Γ(R) whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are joined by an edge iff xy = 0. In this article, we determine precisely those planar graphs that can be realized as Γ(R) when R is an infinite commutative ring.

KW - Zero-divisor graphs

UR - http://www.scopus.com/inward/record.url?scp=33847191050&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847191050&partnerID=8YFLogxK

U2 - 10.1080/00927870601041458

DO - 10.1080/00927870601041458

M3 - Article

VL - 35

SP - 171

EP - 180

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 1

ER -