## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 46, Number 1 (2017), 107-118.

### More on the annihilator graph of a commutative ring

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

#### Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann_R(xy)\neq ann_R(x)\cup ann_R(y)$. In this paper, we study the affinity between annihilator graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the annihilator 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 if and only if $R$ has at most two associated primes. Among other results, under some assumptions, we give necessary and sufficient conditions under which $AG(R)$ is a star graph.

#### Article information

**Source**

Hokkaido Math. J., Volume 46, Number 1 (2017), 107-118.

**Dates**

First available in Project Euclid: 30 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1498788098

**Digital Object Identifier**

doi:10.14492/hokmj/1498788098

**Mathematical Reviews number (MathSciNet)**

MR3677877

**Zentralblatt MATH identifier**

1362.13003

**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**

Annihilator graph Zero-divisor graph Associated prime ideal

#### Citation

NIKMEHR, M. J.; NIKANDISH, R.; BAKHTYIARI, M. More on the annihilator graph of a commutative ring. Hokkaido Math. J. 46 (2017), no. 1, 107--118. doi:10.14492/hokmj/1498788098. https://projecteuclid.org/euclid.hokmj/1498788098