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Let $\mathbb Q$ be the set of all rational numbers. I would like to know what the ideal for $\mathbb Q$ as ring is. I think the ideal of $\mathbb Q$ is $\mathbb Q$, Am I right?

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    See also this question: [A ring is a field iff the only ideals are (0) and (1)](http://math.stackexchange.com/questions/101157/a-ring-is-a-field-iff-the-only-ideals-are-0-and-1)2012-02-01

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An ideal must enjoy the property that if you multiply any of its members by any rational number, what you get is still in the ideal. But if a member $x$ of the supposed ideal is not $0$, then it's easy to show that if you take all products of $x$ with rational numbers, the set that you get, which is $\{xy:y\in\mathbb{Q}\}$, is all of $\mathbb{Q}$. So nothing smaller than $\mathbb{Q}$ can be an ideal in this ring, except $\{0\}$.

Here's the proof: suppose $w$ is any member of $\mathbb{Q}$. Then $w/x$ is rational. So $w/x$ is the value of $y$ that will serve.