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Let $S$ be a subring of a commutative ring $R$ and $a,b,c,d$ are all in $S$. If ideals $(a,b) = (c,d)$ in R, does $(a, b) = (c,d)$ still hold in $S$? If not, what would be a necessary condition on $R$ or $S$ for the statement to hold true?

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No. For example, let $R = \mathbb{Q}, S = \mathbb{Z}$. – Qiaochu Yuan Dec 21 '12 at 10:40

If $S\subset R$ is faithfully flat or $S$ is a direct summand of $R$ (considered as $S$-module), then $IR\cap S=I$ for any ideal $I\subset S$.

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