Rings of square stable range one

A ring S is said to have square stable range one (written ssr(S)=1) if aS+bS=S implies that a2+bx is a unit for some x∈S. In the commutative case, this extends the class of rings of stable range 1, and allows many new examples such as rings of real-valued continuous functions, and real holomorphy rings. On the other hand, ssr(S)=1 sometimes forces S to have stable range 1. For instance, this is the case for exchange rings S, for which ssr(S)=1 is characterized by S/radS being reduced (or abelian, or right quasi-duo). We also characterize rings S whose (von Neumann) regular elements are strongly regular, by using an element-wise notion of square stable range one. Extending a result of Estes and Ohm, we show that a possibly noncommutative infinite domain with stable range one or square stable range one must have a non-artinian group of units.