Let $R$ be a subring of $Q$ containing 1. Then which of the following is/are true?
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$R$ is a PID.
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$R$ contains infinitely many prime ideals.
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$R$ contains a prime ideal which is not a maximal ideal.
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For every maximal ideal $m$ , $R/m$ will be finite.
My Try : 2. This is not true. $Q$ itself is the counter example. $Q$ has no ideal.
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This is correct. Because {0} is in every $R$.
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If we take the subring [$a/3^k$: where k is non negative integer]. Then $Z$ is a maximal ideal of this subring but $R/Z$ is not finite. so False.
I have no idea about option 1.
Have I gone wrong anywhere? Please correct me if I have and tell me what will happen for the option 1.
Best Answer
2, 3, and 4 are obviously false, since $\Bbb{Q}$ is an obvious counterexample in each case.
1 is true since the subrings are various localizations of $\Bbb{Z}$, and localizations of PIDs are PIDs.