# Example of an associative unital ring R with stable range 1 and Jac(R)=0 that is not an exchange ring

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?

## 1 Answer

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.

• Nice! Thanks a lot. Yes, for an exchange ring that is a domain, an element x is either invertible or 1-x is invertible. – Hugo Jan 16 '20 at 21:13