The Jacobson property in rings and Banach algebras

In a Banach algebra A it is well known that the usual spectrum has the following property: sigma(ab) \ {0} = sigma(ba) \ {0} for elements a, b is an element of A. In this note we are interested in subsets of A that have the Jacobson Property, i.e. X subset of A such that for a, b is an element of A: 1 - ab is an element of X double right arrow 1 - ba is an element of X. We are interested in sets with this property in the more general setting of a ring. We also look at the consequences of ideals having this property. We show that there are rings for which the Jacobson radical has this property.