In a PID $R$, given $a,b\in R$, what are the ideals $\langle a\rangle,\langle b\rangle$ also generated by?

Question

Let $R$ be a principal ideal domain and let $a$ and $b$ be two nonunit elements in $R$ then ideal generated by $a$ and $b$ is also generated by

1. $a+b$

2. $ab$

3. $\gcd(a,b)$

4. $\text{lcm}(a,b)$

I am not sure but I think that answer is 3.

Let $I=\langle a,b \rangle$ $=\{ax+by:x,y\in R\}$. Let $d=gcd(a,b)$. Since $R$ is a PID we get $d=ax+by$ for some $x,y\in R$. So $\langle d \rangle \subseteq I$ as $d$ is a divisor of $a$ and $b$ clearly $a,b \in \langle d \rangle $ So $I \subseteq \langle d \rangle$. Hence $I=\langle d \rangle$. Is it correct?

Answer 1

Yes, it is correct. In fact, some authors go a step further and define the gcd of $a$ and $b$ of a ring to be a generator of the smallest principal ideal containing both $a$ and $b$.