Let $I = n\mathbb{Z}$ , $J = m\mathbb{Z}$. Does $I+J = I \cap J = IJ = I/J = d\mathbb{Z}$???

Question

Problem: Let $I = n\mathbb{Z}$ , $J = m\mathbb{Z}$. Find $I+J, I \cap J, IJ, I/J$

My question: If we let $d = \gcd(n,m)$, does $I+J = I \cap J = IJ = I/J = d\mathbb{Z}$???

$a \in I \cap J \Rightarrow$ both $a \in I$ and $a \in J$. $d\mathbb{Z} \in I$ and $d\mathbb{Z} \in J \Rightarrow d\mathbb{Z} = I \cap J$. With $I + J, IJ, I/J$ we have a similar proof.

Anything wrong here? Thank all!

Answer 1

$\mathbb{Z}$ is a principal ideal domain, so the idea of having one element spanning each of the ideals(are all of them ideals?) is correct. But for example the intersection is spanned by the $\text{lcm}(n,m)$ not the $\text{gcd}(n,m)$.

$$x\in I\cap J \Rightarrow (n|x) \wedge (m|x) \Rightarrow l=\text{lcm}(n,m)|x$$ So $I\cap J \subseteq (l)$.

For the other direction suppose $x\in (l)$ but not in either, derive a contradiction.

For $I+J$ consider the $\text{gcd}(n,m)$. For $IJ$ check if its included in one of the previous two.