Is $a+b-1$ a unit in commutative ring?

Question

Let $R$ be a commutative ring and $a,b\in R$ are units.

Can we prove that $a+b-1$ is a unit in $R$?

I think we can prove this with some smart tricks, but I've tried for a long time without making any progress.

Thanks for your hint!

Answer 1

$\Bbb Z$ is a commutative ring. Also $-1$ is a unit in $\Bbb Z$, and so is $-1$. But $(-1)+(-1)-1=-3$

Answer 2

The ring of rational expressions in $b$ with poles only at $b\in\{0,1\}$ $$R=\mathbb{Z}[b,b^{-1},(b-1)^{-1}]$$ has both $b$ and $1-b$ units, but $$b+(1-b)-1=0\text{.}$$

Answer 3

Let $R=\mathbb R$, the field of real numbers. Let $a=b=\frac12$. See that $\frac12+\frac12-1=0$.