This is in regards to definition 6.8, p. 62 from Fraleigh's "A first course in abstract algebra".
6.8 Definition
Let $r$ and $s$ be two positive integers. The positive generator $d$ of the cyclic group $$H=\{nr+ms|n,m\in\mathbb{Z}\}$$ under addition is the greatest common divisor (abbreviated gcd) of $r$ and $s$. We write $d=\gcd(r,s)$
Note from the definition that $d$ is a divisor of both $r$ and $s$ since both $r=1r+0s$ and $s=0r+1s$ are in $H$. Since $d\in H$, we can write $$d=nr+ms$$ for some integer $n$ and $m$. We see that every integer dividing both $r$ and $s$ divides the right hand side of the equation, and hence must be a divisor of $d$ also. Thus $d$ must be the largest number dividing both $r$ and $s$; this accounts for the name given to $d$ in Definition 6.8.
Basically I don't understand this definition at all, and I am hoping someone can break it down for me why it makes sense.
For example:
"$d$ is a divisor of both $r$ and $s$ since both $r=1r+0s$ and $s=0r+1s$ are in $H$"
$d$ is a generator so $\langle d\rangle=H$, but I don't see why $d$ should be a divisor of $r$ and $s$.
Why does $H$ have anything to do with divisors? If you wrote out $\{d^n|n\in\mathbb{Z}\}$ why would it contain divisors for $r,s$? Is every element a divisor, or just some of the elements?
I'd really appreciate if someone could explain the gist of this.
Let $H = r\mathbb{Z} + s\mathbb{Z} $ .
$H$ is a subgroup, i.e. an ideal of $\mathbb{Z}$, which is a PID, and so $H = <d> =d\mathbb{Z}$, i.e. the cyclic group generated by $d$, for some $d \in \mathbb{N}$ .
$r, s \in H$ , so $d \mid r $ and $d \mid s$ by the definition of cyclic group. Moreover if $k \in \mathbb{N}$ is such that $k \mid r $ and $k \mid s$ this means that $r = ka $ and $s = kb $ for $a, b \in \mathbb{Z}$ then $$H = r\mathbb{Z} + s\mathbb{Z} = ka\mathbb{Z} + kb\mathbb{Z}\subseteq k\mathbb{Z}$$ $$H = d\mathbb{Z} \subseteq k\mathbb{Z}$$ and so $k \mid d$ . But this is precisely the "elementary "definition of greatest common divisor, and so $$d = \gcd(r,s)$$