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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Each ideal of a ring is a subring of that ring (see this), therefore no such example can be found.
P..
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1I thought an ideal is a subset of a ring but not subring ? – Idonknow Mar 24 '13 at 06:36
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@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 06:41
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11This depends on if you want your rings to have an identity or not. – anon Mar 24 '13 at 06:47
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the definition of subring: An nonempty subset of ring which closed under subtraction and multiplication – Idonknow Mar 24 '13 at 06:47
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@Idonknow: And the definition of an ideal? – P.. Mar 24 '13 at 06:52
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Ideal: A subset of a ring which closed under addition and for all $r \in R$(ring) , $rI \subset I$ and $Ir \subset I$ – Idonknow Mar 24 '13 at 06:58
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If $a,b\in I$ then $(-1)b=-b\in I\Longrightarrow a+(-b)=a-b\in I$. Therefore $I$ is closed under subtraction. That being said, is my answer correct? – P.. Mar 24 '13 at 07:03
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