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?



Example of an associative unital ring R with stable range 1 and Jac(R)=0 that is not an exchange ring
I know at least two:


1. The integral closure of $\mathbb Z$ in $\mathbb C$
2. The ring of holomorphic functions on $\mathbb C$.


Each nontrivial ideal of an exchange ring with Jacobson radical zero must contain a nonzero idempotent, but since these are both domains, this is clearly not the case for them.


I found these using [this DaRT query](http://ringtheory.herokuapp.com/search/results/?H=27&H=6&L=47).


