How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$?

Question

How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$?

($(2)$ is the principal ideal generated by $2$ over the ring of integers.)

I wrote $(2x)=\{(2x)z:z \in \mathbb{Z}\}=\{2y:y \in \mathbb{Z}\}=(2)$ where $y=zx$.

Does this make sense?

Answer 1

Your statement is not true, simply because $2 \in (2)$ but $2 \notin (2x)$ for $x>1$. If $2$ is in the principal ideal of some element $(x)$, then $x$ divides $2$, so $x$ can only be $1$ ( the whole ring), or $2$ ( which stands for $(2)$).

Yes, it is true that that the ideal $(2x)$ will be contained in the ideal $(2)$, because $2$ divides $2x$, but the converse is not even close to true.