Which of the following is/are true.
- $\mathbb{Z}[x]$ is a principal ideal domain.
- $\mathbb{Z}[x,y]/\langle y+1 \rangle$ is a unique factorization domain.
- If $R$ is a PID and $p$ is a non-zero prime ideal, then $R/p$ has finitely many prime ideals.
- If $R$ is a PID, then any subring of $R$ containing 1 is again a PID.
Option 1 is obviously false. Also for option 3, since $R$ is a PID, so $R/p$ is a field and field has only one prime ideal. So option 3 is true. But I have no clue how to solve option 2 and 4. Can anybody help me with these? Thanks.
Best Answer
You can use (the falsity of) (1) to aid in (4). The ring $\mathbb{Q}[X]$ is a PID, but contains $\mathbb{Z}[X]$ as a subring containing $1$.