ON THE LATTICE OF z ◦ -IDEALS (RESP., z-IDEALS) AND ITS APPLICATIONS

Abstract

An ideal $I$ of a commutative ring $R$ is a $z^{\circ}$-ideal (resp., $z$-ideal) if, for each $a\in I$, the intersection of all minimal prime ideals (resp., maximal ideals) containing $a$ is contained in $I$.
A ring $R$ is termed a $WSA$-ring if, for any two ideals $I, J$ of $R$, where $I\cap J=0$, we have $(\Ann(I)+\Ann(J))_{\circ}=R$. It is observed that for a reduced ring $R$, the lattice of $z^{\circ}$-ideals of $R$ ($Z^{\circ}Id(R)$) is a co-normal lattice \ifif $R$ is a $WSA$-ring. This concept is then applied to characterize spaces $X$ for which $C(X)$ is a $WSA$-ring. In this context, a space $X$ is termed a $WED$-space if every two disjoint open sets can be separated by two disjoint Z-zero-sets (i.e., the interior of a zero-set). The class of $WED$-spaces contains the class of extremally disconnected spaces and the class of perfectly normal spaces. It has been proven that $C(X)$ is a $WSA$-ring \ifif $X$ is a $WED$-spaces, and also \ifif $C^*(X)$ is a $WSA$-ring. Moreover, it has been demonstrated that the lattice of z-ideals of a commutative ring $R$ ($ZId(R)$) is a co-normal lattice \ifif $R$ is an $SA$-ring, and also \ifif the lattice of radical ideals of $R$ ($RId(R)$) is a co-normal lattice.