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