Ideals of Unique Factorization Domain

Question

Let R be a commutative ring with unity such that R[x] is UFD. The ideal (x) of R[x] is denoted by I. Then pick the correct statements from below:
1. I is prime.
2. If I is maximal then R[x] is a PID.
3. If R[x] is a Euclidean domain then I is maximal.
4. If R[x] is a PID, then it is a Euclidean domain.

According to me third is wrong because Z[x] is a Euclidean domain but I is not maximal. What about the other options?

$Z[x]$ is not Euclidean. What do you get when you divide $x$ by 2? — Gerry Myerson, Jun 29 '16 at 07:25
It seems like all four are correct, so the question is a little strange. — rschwieb, Jun 29 '16 at 10:57

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

1

Hint: $R[x]/I \cong R$.
1- $I$ is prime iff $R$ is domain.
2- $I$ is maximal iff $R$ is field.
3- use (2) with the fact: $R$ is a commutative integral ring, $R[X]$ is a principal ideal domain imply $R$ is a field.
4- By the link above, $R$ is a field.

edited Apr 13 '17 at 12:21

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answered Jun 29 '16 at 07:38

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Ideals of Unique Factorization Domain

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