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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. 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