Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas