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Let $R$ be a commutative ring with $1$ and $S $ be a multiplicative subset of $R $. I am looking for an equivalence condition for the following property in $R $:

Property: There exists a fixed element $s\in S $ such that for each idempotent element $e $ of $R $ we have either $se=0$ or $s (1-e)=0$.

Note: Above property has a geometric interpretation, actually, $s $ is an element that annihilates a set of idempotents whose closure is a connected component in the $Spec (R) $, when $s $ is not a unite element.

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    $\begingroup$ The collection of clopen subsets of $\operatorname{Spec}(R)$ on whose complement $s$ vanishes identically forms a filter (it is closed under intersections, enlarging the subset, and doesn't contain the empty set). Your property is equivalent to it being a maximal filter. The usual construction using Zorn's lemma should show that such an element always exists. $\endgroup$
    – Bertram Arnold
    Commented Jan 21, 2020 at 15:04
  • $\begingroup$ Thanks for your comment. I fixed the question. $\endgroup$
    – Artur
    Commented Jan 21, 2020 at 15:33
  • $\begingroup$ Your comment is useful for me. $\endgroup$
    – Artur
    Commented Jan 21, 2020 at 15:33
  • $\begingroup$ Can you explain a little more about filters? $\endgroup$
    – Artur
    Commented Jan 21, 2020 at 15:37

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