Intersections of essential minimal prime ideals

Let \(\mathcal{Z(R)}\) be the set of zero divisor elements of a commutative ring \(R\) with identity and \(\mathcal{M}\) be the space of minimal prime ideals of \(R\) with Zariski topology. An ideal \(I\) of \(R\) is called strongly dense ideal or briefly \(sd\)-ideal if \(I\subseteq \mathcal{Z(R)}\) and is contained in no minimal prime ideal. We denote by \(R_{K}(\mathcal{M})\), the set of all \(a\in R\) for which \(\bar{D(a)}=\bar{\mathcal{M}\setminus V(a)}\) is compact. We show that \(R\) has property \((A)\) and \(\mathcal{M}\) is compact \ifif \(R\) has no \(sd\)-ideal. It is proved that \(R_{K}(\mathcal{M})\) is an essential ideal (resp., \(sd\)-ideal) \ifif \(\mathcal{M}\) is an almost locally compact (resp., \(\mathcal{M}\) is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring \(R\) need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring \(R\) is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of \(C(X)\) is equal to the socle of C(X) (i.e., \(C_{F}(X)=O^{\beta X\setminus I(X)}\)). Finally, we show that a topological space \(X\) is pseudo-discrete \ifif \(I(X)=X_{L}\) and \(C_{K}(X)\) is a pure ideal.