Intersections of essential minimal prime ideals

Taherifar, A.

Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 1, pp. 121-130.

Résumé

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 $I$ is contained in no minimal prime ideal. We denote by $R_{K}(\mathcal{M})$, the set of all $a\in R$ for which $\overline{D(a)}= \overline{\mathcal{M}\setminus V(a)}$ is compact. We show that $R$ has property $(A)$ and $\mathcal{M}$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_{K}(\mathcal{M})$ is an essential ideal (resp., $sd$-ideal) if and only if $\mathcal{M}$ 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 if and only if $I(X)=X_{L}$ and $C_{K}(X)$ is a pure ideal.

Détail

Classification : 13A15, 54C40
Keywords: essential ideals; $sd$-ideal; almost locally compact space; nowhere dense; Zariski topology