True/False regarding PIDs and UFDs

Question

The question is as follows:

Which of the following statements are true?

$\Bbb Z[x]$ is a PID

$\Bbb Z[x,y]/\langle y+1 \rangle$ is a UFD

If $R$ is a PID and $\mathfrak p$ is a non-zero prime ideal, then $R/\mathfrak p$ has finitely many prime ideals

If $R$ is a PID, then any subring of $R$ containing $1$ is again a PID.

(PID = principal ideal domain, UFD = unique factorization domain)

please somebody help me with this question i know the first option is incorrect because Z is not a field but the rest i dont know please help

PLEASE DO NOT TYPE IN CAPS. It makes it seem as though you're shouting. — Ben Grossmann, Dec 19 '16 at 18:56
https://math.stackexchange.com/q/2200723/279515 is the same question. — , Jun 06 '18 at 21:28

Bernard · Answer 1 · 2016-12-19T22:10:54.923

2

Hints:

2) $\mathbf Z[x,y]/(y+1)$ is isomorphic to .....

3) What can you say about a non-zero prime ideal in a P.I.D.?

4) The subring of $\mathbf Q[x]$ of integer-valued polynomials is not noetherian (not sure this is the simplest possible answer for beginners…).

4) Simpler (thanks to @Omnomnomnom): See in question 1: $\;\mathbf Z[x]\subset \mathbf Q[x]$

edited Dec 19 '16 at 22:10

answered Dec 19 '16 at 19:00

Bernard

179,256

For 4: question 1 already asks about $\Bbb Z[x]$ so there's no need to appeal to ascending chains. – Ben Grossmann Dec 19 '16 at 19:07
Quite right. I'll add it as a hint (with due credits of course). – Bernard Dec 19 '16 at 19:08
@Bernard Z[x,y]/(y+1) is isomorphic to which group – Inverse Problem May 14 '17 at 09:02
@Bernard.. a non zero prime ideal in PID is maxiaml – Inverse Problem May 14 '17 at 09:08
@suresh: working in $\mathbf zx,y]/(y+1)$ amounts to replace $y$ by $-1$ in a polynomial $p(x,y)$. So you obtain the polynomial $p(x, -1)\in\mathbf Z[x]$. – Bernard May 14 '17 at 10:53

score 0 · Answer 2 · answered Dec 19 '16 at 19:07

0

Hints:

1.) The Krull dimension of $\mathbb{Z}[x]$ is $\dim \mathbb{Z}+1$, and PID's have Krull dimension $0$ or $1$ (not sure this is the simplest possible answer for beginners, but elementary answers are already duplicates).

2.),3.),4.) Bernard.

answered Dec 19 '16 at 19:07

Dietrich Burde

140,055

True/False regarding PIDs and UFDs

2 Answers2