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Give an example of an ideal that is not a subring, and a subring that is not an ideal. For the latter part, let $\mathbb{Q}$ be a ring and consider $\mathbb{Z}$ as a subring of $\mathbb{Q}$. Then we observe that $\mathbb{Z}$ is not an ideal of $\mathbb{Q}$ since the multiplication of integer and rational number gives rational number. But for the first part, I have no idea. Can anyone guide me ?

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up vote 3 down vote accepted

Each ideal of a ring is a subring of that ring (see this), therefore no such example can be found.

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I thought an ideal is a subset of a ring but not subring ? –  Idonknow Mar 24 '13 at 6:36
    
@Idonknow: Can you give the definition of a subring please. Maybe my answer is wrong. It depends on what definition you use. –  P.. Mar 24 '13 at 6:41
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This depends on if you want your rings to have an identity or not. –  anon Mar 24 '13 at 6:47
    
the definition of subring: An nonempty subset of ring which closed under subtraction and multiplication –  Idonknow Mar 24 '13 at 6:47
    
@Idonknow: And the definition of an ideal? –  P.. Mar 24 '13 at 6:52

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