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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: σ ( a b ) \ { 0 } = σ ( b a ) \ { 0 } for elements a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. X ⊂ A such that for a , b ∈ A : 1 - a b ∈ X ⇒ 1 - b a ∈ X . W...
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| Published in: | Afrika mathematica 2023-09, Vol.34 (3), Article 50 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | In a Banach algebra
A
it is well known that the usual spectrum has the following property:
σ
(
a
b
)
\
{
0
}
=
σ
(
b
a
)
\
{
0
}
for elements
a
,
b
∈
A
. In this note we are interested in subsets of
A
that have the Jacobson Property, i.e.
X
⊂
A
such that for
a
,
b
∈
A
:
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. |
|---|---|
| ISSN: | 1012-9405 2190-7668 |
| DOI: | 10.1007/s13370-023-01093-1 |