Rings are supposed to be associative and unital, but not necessarily commutative. Some definitions:
- (Bass) A ring $R$ is said to have stable range $1$ if for all $a,b \in R$, whenever $Ra+Rb=R$, then there exists $x\in R$ with $a+xb$ being a unit.
- (Warfield) A ring $R$ is said to be an exchange ring if it has the finite exchange property or equivalently (Nicholson) if for all $a\in R$ there exists an idempotent $e\in R$ with $e\in Ra$ and $1-e \in R(1-a)$.
Examples: Semiperfect rings have stable range $1$ and are exchange.
My question is whether someone has an example of a ring $R$ with stable range $1$ and Jacobson radical zero, that is not exchange?