Is the reason that $\displaystyle n\mathbb{Z} \not\cong m\mathbb{Z}$ in general because if we define a map of the type $\displaystyle x \mapsto ax$, the inverse $\displaystyle ax \mapsto \frac{1}{a}ax$ doesn't exist, since $\displaystyle \dfrac{1}{a}$ isn't in $\displaystyle n\mathbb{Z}$ or $\displaystyle m\mathbb{Z}$? (the text doesn't require a ring to have unity)

I have a follow-up question depending on the answer. Thanks.

(by the way, I am new to this forum and read the rules, but if there's anything I should be aware of, please let me know)