If $1-x$ is a ring unit then is this ring nilpotent?

Question

It can be shown that if $R$ is a commutative unital ring and $x$ is nilpotent, then $1-x$ is a unit of $R$. But is the converse true? I'm not sure how to approach this problem. That is, is it true that if $R$ is a commutative unital ring and $1-x$ is a unit, then $x$ is nilpotent?

Answer 1

Seems not.

Take the field (thus commutative ring) of real numbers, and take $x$ to be any real number other than 1. Then $1-x$ is certainly a unit, but $x$ is not nilpotent.

Answer 2

To get an instant counterexample, take any commutative unital ring $R$ such that

• $R$ has no nonzero nilpotent elements.
• $R$ has at least two units.

Now let $x = 1 - u$, where $u$ is a unit other than $1$.

Then $1 - x$ is a unit, but since $u \ne 1$, $x$ is nonzero, hence by choice of $R$, $x$ is not nilpotent.

For example, let $R$ be any integral domain such that $-1 \ne 1$ (i.e., $R$ does not have characteristic $2$), and let $x = 2$.