I'm trying to teach myself by doing questions. I understand the definition of a ideal is a multiplicatively closed additive subgroup of a ring. And a principal ideal means it has a generator 'g'. So from this I guess the integers have a principal ideal domain. Where I'm stuck is the three polynomial examples and the reasons why so I can learn and apply in other circumstances.
Edit: For part a, I'm really looking for like a rule of thumb of picking out principal ideal domains if there is one.
As for part b, I really do have no idea and just looking for general help.
Very grateful, thank you.
For the polynomial examples, note that in $\mathbb{Z}[X,Y]$ the ideal $(X,Y)$ isn't principal.
Also, $\mathbb{Z}_{2011}$ is a field, and $F[X]$ is a PID if and only if $F$ is a field.
For b), note that $(a) \cap (b)$ is generated by the single element $d$, where $d$= g.c.d of $a$ and $b$.