In an UFD which of the following ideals is prime?

Let $R$ be a unique factorization domain and let $a,b\in R$ be distinct irreducible elements. Could anyone tell me which of the following is true?

1. $\langle 1+a\rangle$ is a prime ideal.

2. $\langle a+b\rangle$ is a prime ideal.

3. $\langle 1+ab\rangle$ is a prime ideal

4. $\langle a\rangle$ is not necessarily a maximal ideal.

I remember the definition of irreducible element: an element $f\in R$ such that there does not exist non-units $g,h$ such that $f=gh$, and in a UFD, prime and irreducible elements coincide. $4$ is prime ideal right?

• It is easy to find counterexamples to the first three by just looking at $\mathbb{Z}$. – Tobias Kildetoft Aug 2 '13 at 11:52
• you're right that $4$ will be prime, so think about examples of prime ideals that aren't maximal – citedcorpse Aug 2 '13 at 11:53
• < and > mean "less than" and "greater than", and produce spacing correct for that meaning only; to make angle brackets, use \langle and \rangle. – Zev Chonoles Aug 2 '13 at 11:53
• @exitingcorpse (and Taxi Driver): $4$ is not prime... – Zev Chonoles Aug 2 '13 at 12:06
• @ZevChonoles I'm pretty sure that exitingcorpse meant point 4., $\langle a\rangle$ will be a prime ideal (trivially). – Daniel Fischer Aug 2 '13 at 12:10

For 1., Take $R=\mathbb{Z}$, then $5$ is irreducible in $\mathbb{Z}$ but $\langle 1+5 \rangle = \langle 6\rangle$ is not a prime ideal.
For 2., set $a=5$ and $b=3$, both are irreducible but $\langle 8\rangle$ is not a prime ideal of $\mathbb{Z}$
For 3., again set $a=5$ and $b=3$.
For 4., since a is irreducible $\langle a \rangle$ is a prime ideal, is it maximal? well, not necessarily. Take $\langle x^2+y \rangle$ in $\mathbb{R}[x,y]$, it's a prime ideal since $x^2+y$ is irreducible, but it's not maximal since $\langle x^2+y \rangle \subset \langle x,y \rangle$.
• or even more simply, take $\langle x \rangle$. certainly $\mathbb{R}[x, y]/ \langle x \rangle = \mathbb{R}[y]$ is an integral domain, so the ideal is prime, but far from maximal (any of the ideals $\langle x, y^n \rangle$ will contain it) – citedcorpse Aug 2 '13 at 12:47