Let $(R,+,\times)$ be a finite ring. $R^\times$ denotes the set of all invertible elements, i.e., units in $(R,\times)$.
Find finite rings $(R,+,\times)$ such that for every unit $r\in R^\times\setminus\{1\}$, $r-1$ is a unit.
I know that finite fields $\mathbb{F}_q$ are such rings. Then I try to prove that they must be finite field.
My Attempt:
Assume that $r$ and $r-1$ are units. Then there exist $x,y\in R^\times$ such that $$rx=xr=1$$ and $$(r-1)y=y(r-1)=1.$$ Now we have $$(r-rx)y=(r-1)y=1=rx$$ and thus $$(1-x)y=x.$$
I do not know how to contiue...
Boolean Algebras B are "such rings".
Indeed, they have only one unit, $\ B^x=\{1\}.$