Abstract
AbstractIn 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.
Funder
University of Johannesburg
Publisher
Springer Science and Business Media LLC
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