Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with identity, $X$ the set of all nonzero, nonunits of $R$ and $G$ the group of all units of $R$. We will investigate some ring theoretic properties of $R$ by considering $\Gamma (R)$, the zero-divisor graph of $R$, under the regular action on $X$ by $G$ as follows: (1) If $R$ is a ring such that $X$ is a union of a finite number of orbits under the regular action on $X$ by $G$, then there is a vertex of $\Gamma (R)$ which is adjacent to every other vertex in $\Gamma (R)$ if and only if $R$ is a local ring or $R \simeq \Bbb Z_{2} \times F$ where $F$ is a field; (2) If $R$ is a local ring such that $X$ is a union of $n$ distinct orbits under the regular action of $G$ on $X$, then all ideals of $R$ consist of $\{\{0\}, J, J^{2}, \dots, J^{n}, R\}$ where $J$ is the Jacobson radical of $R$; (3) If $R$ is a ring such that $X$ is a union of a finite number of orbits under the regular action on $X$ by $G$, then the number of all ideals is finite and is greater than equal to the number of orbits.

Keywords: zero-divisor graph, regular action, orbit, local ring

MSC numbers: Primary 13H99; Secondary 16E50