which of the following statements are true and why?

Question

which of the following statements are true and why?

1. Any two irreducibles in any UFD are associates.

2. If $D$ is a PID, then $D[x]$ is a PID.

3. In any UFD, if $p|a$ for an irreducible $p$, then $p$ itself appears in every factorization of $a$.

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can anyone help me to find the proofs and counter examples of the above as reasons.thanks for your time.

Answer 1

Hint 1) Look at $\Bbb Z$.

Hint 2) Look at $D=\Bbb Z$.

Hint 3) Let $u$ be any unit besides 1, and $q=pu$ is also prime. Now $p|q$. Does $p$ appear in the factorization of $q$ as just $q$? (I am not totally sure what counts as a factorization to you, so this is the best hint I can think of!)