If the group of units of a unital ring is cyclic, must it be finite?

Question

Suppose you have a ring $R$ with $1$ and that the multiplicative subgroup $R^\times$ of $R$ is cyclic.

Is then $R^\times$ a finite group?

Are there conditions, such that $R^\times$ is cyclic, or is it and if then else relation between these two properties.

EDIT: Oky it is wrong! Is this true in Char $R\neq 2,1$?

Answer 1

The ring $R=\mathbb{F}_2[x,x^{-1}]$ is a counterexample (not the only one; see the comments), with $$R^\times=\{\ldots,x^{-1},1,x,\ldots\}$$ forming an infinite cyclic group.