A PID that is a sub ring of a Non-PID

Question

I’ve read a post asking whether a subring of a PID is always a PID. The answer is no, but the post itself gave me more questions.

1. Is that possible for a PID that is a subring of a non-PID?

2. Is that possible for a subring of a PID that is not a UFD?

Some hints or examples are really appreciated!

Thank you!

Answer 1

The answer to your first question is yes, as in the comments, $\Bbb Z$ is a subring of $\Bbb Z[X]$, which is a PID, but $\Bbb Z[X]$ is not a PID (look at the ideal $(2, X)$ for the canonical example).

For your second question, pick your favourite ring which is not a unique factorisation domain and then extend to its field of fractions. For example, $\Bbb Z[\sqrt{-5}]$ is not a unique factorisation domain, while $\Bbb Q(\sqrt{-5})$ is a principal ideal domain (in particular, a field).

Answer 2

1. $\Bbb Z\times \Bbb Z$ has $\Bbb Z\times 0\cong \Bbb Z$ as a subring.
2. Every PID is a UFD