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In general the union of chain of ideals of a ring is an ideal. But I did not get the following idea.

Let $I$ be an ideal of an integral domain $R$ and let $J = \bigcup_{A\subseteq I} (A^{-1})^{-1},$ where $A$ is a finitely generated ideal of $R$ contained in $I$, $A^{-1} = \langle{x: x \in K, xA \subseteq R}\rangle $ and $K$ is the quotient field of $R$. Then $J$ is an ideal of $R$.

This appear in closure operation of set of ideals. Thanks in advance.

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    Every finite subset of $I$ is contained in a finitely generated ideal of $R$ which is contained in $I$, so that union is just $I$, no? – 2017-02-19
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    Sorry I just correct the question. – 2017-02-19
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    Isn't this trivial? At least show us that if $r\in R$ and $x\in J$ then $rx\in J$. – 2017-02-19
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    Thank you user 26857. It seems the addition is not obvious. – 2017-02-19
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    Well, for addition chose the sum of the two finitely generated ideals. – 2017-02-19

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