ring of polynomials and factorization

Question

These are standard facts:

1. $R$ field implies $R[x]$ is a Euclidean domain

2. $R$ is a UFD implies $R[x]$ is a UFD

3. $R$ is an integral domain if and only if $R[x]$ is an integral domain

My questions are:

1. If $R$ is a factorization domain is $R[x]$ a factorization domain (and what about the converse)?

2. If $R$ is a PID is $R[x]$ a PID (and what about the converse)?

Answer 1

You're asking 4 questions. If instead of factorization domain in question 1) you mean Euclidean domain, then this answers both 1) and 2), but not the converses:

$\mathbb{R}[X]$ is a Euclidean domain, but $\mathbb{R}[X][Y]\cong \mathbb{R}[X,Y]$ is not a PID (can you see why?).

Answer 2

Ideas for 1: (a) given a reducible polynomial with no nonunit content, any polynomial factor must be nonconstant hence factoring reduces the degree of the polynomial, (b) if the ring of scalars forms a FD, then we can factor out the content of any polynomial (when it exists).

Claim for 2: given $R$, a domain, $~R[x]$ is a PID $\iff R$ is a field. One direction should be clear using the division algorithm, but what happens if $R$ has a nonunit? Perhaps give an example of a nonprincipal ideal of $\Bbb Z[x]$ in order to get inspiration.