Suppose $R$ is a UFD and that $a,b \in R$ are distinct irreducible elements. Which of the following statements is true?
- The ideal $\langle1+a \rangle$ is a prime ideal.
- The ideal $\langle a+b\rangle$ is a prime ideal.
- The ideal $\langle 1+ab\rangle$ is a prime ideal.
- The ideal $\langle a\rangle$ is not necessarily a maximal ideal.
Since $R$ is an UFD, and $a,b$ are irreducible in $R$ so they are prime elements. So the ideal $\langle a\rangle$ is a prime ideal, but it is not a maximal ideal, as ideal generated by a irreducible element is maximal iff the domain is a PID. So option 4 is correct. But I don't know how to refute the other three options. Any help or hint would be nice. Thank you.
Best Answer
I assume by $<x>$ you mean the principal ideal generated by $x\in R$.
Consider $R=\mathbb{Z}$ and let $a=3$,$b=5$,then $a$ and $b$ are distinct irreducibles.
1.$(1+a)=(4)$ is not prime.
2.$(a+b)=(8)$ is not prime
3.$(1+ab)=(1+15)=(16)$ is not prime.
For 4, you are right. An example is $R=\mathbb{R}[x][y]$, and let $a=x$.