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Let $R$ be a commutative ring with an identity, which contains $\Bbb Z$, the integer field, as a sub-ring with the same identity element.
Let $I$ be a maximal ideal in $R$.
I need to prove that if $\Bbb Z/(I \cap \Bbb Z)$ it not a field, then $R/I$ contains a sub-field that is isomorphic to $\Bbb Q$, the rational field.

I can prove that $\Bbb Z/(I \cap \Bbb Z)$ is infinite, but I don't know how to continue on.

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  • $\begingroup$ It is better to write $\mathbb{Z}/(I\cap \mathbb{Z})$. $\endgroup$ – Dietrich Burde May 16 '14 at 17:59
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You can write $I\cap\mathbb{Z}=n\mathbb{Z}$, with $n\ge0$. Since $$ \frac{\mathbb{Z}}{I\cap\mathbb{Z}}\cong\frac{\mathbb{Z}+I}{I} $$ embeds in $R/I$ which is a domain, this tells you something about $n$. Next, the hypothesis that $\mathbb{Z}/n\mathbb{Z}$ is not a field tells you precisely what $n$ is.

Conclude something about the characteristic of $R/I$, which is a field, so it contains a minimum subfield.

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