What is $\langle a \rangle + \langle b \rangle$ where $a$ and $b$ are natural numbers?

Question

Let $\langle a \rangle$ and $\langle b \rangle$ be ideals in $\mathbb{Z}$, where a and b are natural numbers.

Define $$S = \langle a \rangle + \langle b \rangle = \{x + y \mid x \in \langle a \rangle \text{ and } y \in \langle b \rangle\}.$$

What is $\langle a \rangle + \langle b \rangle$ ?

I know $S \supseteq d\mathbb{Z}$ where $d = \gcd(a,b)$ since there exist integers $m$ and $n$ such that $d = am + bn$. Then any integer multiple of $d$ can be expressed as the sum of an integer multiple of $a$ and $b$.

Answer 1

For the other direction, let $x+y\in S$ where $s\in \langle a\rangle, y\in \langle b \rangle$. Then there are $h,k\in\mathbb{Z}$ such that $x = ha$, $y = kb$.

Then with $d = \gcd(a,b)$, $d$ is a divisor of $a$ and $b$, so there are $n$ and $m$ such that $a = nd$ and $b = md$. So $x + y = ha + kb = hnd + kmd = (hn + km)d \in d\mathbb{Z}$.