Edited :
I have this particular question in abstract algebra assignment given to me.
I have been studying algebra from Thomas Hungerford as a textbook.
Question : Let R be a subring of $\mathbb{Q}$ containing 1 . Then which 1 of following is nessesary true.
A. R is Principal ideal Domain.
B. R contains infinitely many prime ideals.
C. R contains a prime ideal which is a maximal ideal.
D. for every maximal ideal m in R, the residue field R/m is finite.
Attempt : I don't think ring is PID as it need not have a single element which will generate it.
Unfortunately , for B, C, D I am clueless on how can they be approached.
I understand one should give his attempt but I am unable to think anything about B,C and D.
Any hints please!!
Best Answer
B doesn't necessarily hold - take $R = \mathbb{Q}$.
D also doesn't necessarily hold - again, take $R = \mathbb{Q}$.
It seems like both A and C hold.
A - let $w$ be an ideal of $R$. Let $k = \mathbb{Z} \cap w$. Then we see that $k$ is an ideal of $\mathbb{Z}$. Take $n \in k$ s.t. $k = (n)$ since $\mathbb{Z}$ is a PID. Then $n$ generates $w$. For if we have fully simplified $a/b$ in $w$, then clearly $a \in k$ and therefore $a$ is a multiple of $n$. And since $a$ and $b$ are relatively prime, we can take $x, y \in \mathbb{Z}$ such that $xa + yb = 1$. Then $x (a/b) + y = 1/b$. Then $1/b \in R$. Then $a (1/b)$ is a multiple of $n$. Alternately, since $n \in k$, we have $n \in w$ and thus every multiple of $n$ is in $w$. Then $R$ is a PID.
C - every nonzero ring has a maximal ideal, and every maximal ideal is prime. So $R$ has a maximal, prime ideal.