TITLE: A commutative ring with group of invertible elements isomorphic to $\mathbb{Z}$ has characteristic 2
QUESTION [5 upvotes]: Let $R$ be a commutative ring. Denote by $R^*$ the group of invertible elements (this is a group w.r.t multiplication.) Suppose $R^*\cong  \mathbb{Z}$. I need to show that $1+1=0$ in $R$.

I have no clue about why such statement should be true. I don't even have an example for a ring that satisfies these assumptions, so I'd be glad to see one.
Hints (or partial solutions) will be welcomed.
Thank you!

REPLY [4 votes]: Hint: If $R^{\times}\cong{\mathbb Z}$, then in particular it has no nontrivial torsion; on the other hand, there's $-1\in R^{\times}$.