Rings satisfying *-property

TL;DR

This paper studies commutative rings with the *-property, showing that integral domains with this property are fields, and exploring their zero-dimensionality and relation to Artinian rings.

Contribution

It introduces the *-property for rings, proves that integral domains with this property are fields, and investigates their dimensionality and connection to Artinian rings.

Findings

Integral domains with *-property are fields.

Rings with *-property are zero-dimensional.

Relations between *-property rings and Artinian rings are established.

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.