The Jacobson property in rings and Banach algebras

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