An integral domain R such that every non-zero prime ideal of R[X] is maximal

Question

The following question was asked in my abstract algebra quiz and I had no idea on how to solve it .

Question: Let R be an integral domain such that every non-zero prime ideal of R[X] (where R is an indeterminate) is maximal . Choose the correct option(s) :

1. R is a field

2. R contains $\mathbb{Z}$ as a subring

3. Every ideal in R[X] is principal

4. R contains $\mathbb{F}_p$ as a subring for some prime number p.

Answer :

A, C

I am absolutely clueless on how I should I contract option or how to prove any of them . I tried it 10 days earlier also and there was nothing which I can think of . So , its my humble request to all to please tell me on how to approach this question .

Thanks!!

Answer 1

First, note that if $R$ is an algebraically closed field of arbitrary characteristic, then it is clearly true. This rules out 2 and 4.

1 implies 3, because the ring of polynomials over a field is PID.

If $R$ is not a field, then it has a nonzero proper prime ideal $\mathfrak p$ and $\mathfrak p[x]$ is a prime in $R[x]$, but not maximal.