$x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$

Question

How do I show that $x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$, but that there is no $u\in R^*$ such that $y=ux$? Working with ideals is relatively new to me, so I would appreciate it if somebody could show me the proof for this.

Answer 1

Both 1 and 2 are generators of $\mathbb{Z}/5\mathbb{Z}$. That should be enough for first part.

And as mentioned in comment by @AmitaiYuval, second statement is not true since $(1,3)(0,2)=(0,1)$.