Find all ideals of $\mathbb Q$ 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?

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.
