I need a little help with the second half of the proof that $ (\mathbb{Z} \times \mathbb{Z},+) / \langle(2,3)\rangle $ is cyclic.
I know isomorphism preserves cyclic structure.I looked up the answer on my textbook from where I got this exercise and their proof is like this:
"The function $f : \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} /\langle(2,3)\rangle$ with $f(x) = \widehat{(x,x)}$ is an isomorphism of groups because :
- $(x,x)$ is not in $\langle(2,3)\rangle$ for every $x \neq 0$.
- $(a,b) =(b-a)(2,3)+(3a-2b)(1,1)$ for a,b in $\mathbb{Z}$ .
I think in the first part they try to show that the function is injective but I don't understand the second part,can someone explain it to me? Thanks alot !