Let $R$ be a commutative ring with unity. Which of the following is true:
- If $R$ has finitely many prime ideals, then $R$ is a field.
- If $R$ has finitely many ideals, then $R$ is finite.
- If $R$ is a PID, then every subring of $R$ with unity is a PID.
- If $R$ is an integral domain which has finitely many ideals, then $R$ is a field.
The solution i tried-
We take the example of $\mathbb Z_{2210}$ the prime ideals of this ring are prime divisors of 2210 which are 2,3,7,11,5, but the given ring is not a field, thus we can discard this option.
If we take $R=\mathbb{Q}$, then this is a field (which is also a ring), the only ideals of $\mathbb{Q}$ are $(0)$ and $(1)$, but set of rational number is not finite, so we can discard 2nd option.
In this option we can take $R=\mathbb{Q}[x]$ which is a P.I.D. and take its subring $\mathbb{Z}[x]$, and this subring is not P.I.D., so 3rd option is discarded.
The remained option is 4th which should be true (for this I can't find the example).
I am not satisfied by my approach to this question by only choosing particular examples.
Please suggest me a proper solution, and i am also confused with term "finitely many".
Please help.
Thank you.