Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
In a Banach algebra A it is well known that the usual spectrum has the following property: $$\begin{aligned} \sigma (ab) \setminus \{0\} = \sigma (ba) \setminus \{0\} \end{aligned}$$ σ ( a b ) \ { 0 } = σ ( b a ) \ { 0 } for elements $$a, b \in A$$ a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. $$X \subset A$$ X ⊂ A such that for $$a, b \in A$$ a , b ∈ A : $$\begin{aligned} 1 - ab \in X \implies 1 - ba \in X. \end{aligned}$$ 1 - a b ∈ X ⇒ 1 - b a ∈ 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.
In a Banach algebra A it is well known that the usual spectrum has the following property: $$\begin{aligned} \sigma (ab) \setminus \{0\} = \sigma (ba) \setminus \{0\} \end{aligned}$$ σ ( a b ) \ { 0 } = σ ( b a ) \ { 0 } for elements $$a, b \in A$$ a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. $$X \subset A$$ X ⊂ A such that for $$a, b \in A$$ a , b ∈ A : $$\begin{aligned} 1 - ab \in X \implies 1 - ba \in X. \end{aligned}$$ 1 - a b ∈ X ⇒ 1 - b a ∈ 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.