The Jacobson property in rings and Banach algebras. (English) Zbl 1538.46067
Summary: In a Banach algebra \(A\) it is well known that the usual spectrum has the following property:
\[
\sigma(ab)\setminus\{0\} = \sigma(ba)\setminus\{0\}
\]
for elements \(a, b \in A\). In this note we are interested in subsets of \(A\) that have the Jacobson property, i.e., \(X \subset A\) such that for \(a, b \in A\):
\[
1 - ab \in X \Longrightarrow 1 - ba \in X.
\]
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.
References:
| [1] | Alghazzawi, D.; Leroy, A., Commutatively closed sets in rings, Comm. Algebra, 47, 4, 1629-1641 (2019) · Zbl 1472.16038 · doi:10.1080/00927872.2018.1513011 |
| [2] | Aupetit, B., A primer on spectral theory, Universitext (1991), New York: Springer, New York · Zbl 0715.46023 · doi:10.1007/978-1-4612-3048-9 |
| [3] | Brown, W., Matrices over commutative rings (1992), New York: Marcel Dekker, New York |
| [4] | Cohn, P., Reversible rings, Bull. Lond. Math. Soc., 31, 641-648 (1999) · Zbl 1021.16019 · doi:10.1112/S0024609399006116 |
| [5] | Cvetković-Ilić, D.; Harte, RE, The spectral topology in rings, Stud. Math., 200, 3, 267-278 (2010) · Zbl 1221.46051 · doi:10.4064/sm200-3-4 |
| [6] | Fong, C.-K., Sołtysiak, A.: Existence of a multiplicative functional and joint spectra, Studia Mathematica 81, (1985) · Zbl 0529.46034 |
| [7] | Fong, C.-K., Sołtysiak, A.: On the left and right joint spectra in Banach algebras, Studia Mathematica 97(2), (1990) · Zbl 0737.46039 |
| [8] | Lam, T.Y.: A first course in noncommutative rings, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, (2001) · Zbl 0980.16001 |
| [9] | Raubenheimer, H.; Swartz, A., The Jacobson Property in Banach algebras, Afr. Mat., 33, 36 (2022) · Zbl 1499.46094 · doi:10.1007/s13370-022-00969-y |
| [10] | Warner, S., Topological rings (1993), Amsterdam: Elsevier Science Publishers B.V, Amsterdam · Zbl 0785.13008 |
| [11] | Zemanek J.: Spectral radius characterizations of commutativity in Banach algebras., Studia Matematika, 61, (1977) · Zbl 0321.46037 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
