Rings satisfying *-property

Abstract

In this paper we will investigate commutative rings which have the $\ast$-property. We say that a ring $R$ satisfy $\ast -$property if for any family of ideals ${\left\{I_{\alpha }\right\}}_{\alpha \in S}$ of $R$ in which $S$ is an index set, there exists a finite subset\ $S^{\prime }$ of $S$ such that the radical of the intersection of the family of ideals ${\left\{I_{\alpha }\right\}}_{\alpha \in S}$ is equal to the intersection of the radicals of ideals ${\left\{I_{\alpha }\right\}}_{\alpha \in S^{\prime }}$ . We will show that any integral domain which satisfy $\ast -$property is a field. Furthermore, these rings are zero-dimensional. After this we give relations between these rings and Artinian rings.