## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 46, Number 3 (2017), 381-393.

### The extended zero-divisor graph of a commutative ring I

M. BAKHTYIARI, M. J. NIKMEHR, and R. NIKANDISH

#### Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. It follows that the zero-divisor graph $\Gamma(R)$ is a subgraph of $\Gamma'(R)$. It is proved that $\Gamma'(R)$ is connected with diameter at most two and with girth at most four, if $\Gamma'(R)$ contains a cycle. Moreover, we characterize all rings whose extended zero-divisor graphs are complete or star. Furthermore, we study the affinity between extended zero-divisor graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the extended zero-divisor graph and the zero-divisor graph of $R$ are identical to the join of a complete graph and a null graph if and only if $ann_R(Z(R))$ is a prime ideal.

#### Article information

**Source**

Hokkaido Math. J., Volume 46, Number 3 (2017), 381-393.

**Dates**

First available in Project Euclid: 7 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1510045303

**Digital Object Identifier**

doi:10.14492/hokmj/1510045303

**Mathematical Reviews number (MathSciNet)**

MR3720334

**Zentralblatt MATH identifier**

06814868

**Subjects**

Primary: 13A15: Ideals; multiplicative ideal theory 13B99: None of the above, but in this section 05C99: None of the above, but in this section

**Keywords**

Extended zero-divisor graph Zero-divisor graph Complete graph

#### Citation

BAKHTYIARI, M.; NIKMEHR, M. J.; NIKANDISH, R. The extended zero-divisor graph of a commutative ring I. Hokkaido Math. J. 46 (2017), no. 3, 381--393. doi:10.14492/hokmj/1510045303. https://projecteuclid.org/euclid.hokmj/1510045303