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I'm trying to show by an example that a field $F$ of quotients of a proper subdomain $A$ of an integral domain $D$ may also be a field of quotients of $D$. I have no idea where to begin. Help?

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Instead of starting with $D$, start somewhere between $D$ and $F$. –  Arturo Magidin Nov 28 '11 at 4:48
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Let $B$ be a domain which is not a field, and let $K$ be its field of quotients.

To get your example, pick $D=K$, so that $F=K$, and $A=B$.

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What about D = $\mathbb{Z}$ F=$\mathbb{Q}$ and A=$2\mathbb{Z}$? –  pigishpig Nov 28 '11 at 20:22
    
In this context one usually one considers subrings which contain the same $1$. apart from that: what about that? –  Mariano Suárez-Alvarez Nov 28 '11 at 20:40
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HINT $\ $ Any ring between a domain and its fraction field necessarily has the same fraction field. For some nontrivial examples consider $\rm\:\mathbb Z\subset$ dyadic $\mathbb Q$ $\rm = \{\: m/2^n\ :\ m\in \mathbb Z,\ n\in \mathbb N\:\}\ $ and consider $\rm\:\mathbb Z\subset$ $\mathbb Z_{(2)}$ $\rm = \{\: m/n\ :\ m\in \mathbb Z,\ odd\ n\in \mathbb Z\:\}\:.$

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