The Jacobson radical of an evolution algebra

• Published in: arXiv.org 2018-05
• Main Author: Velasco, M Victoria
• Format: Article
• Language: English
• Subjects: Algebra; Eigenvalues; Evolution; Homomorphisms
• ISSN: 2331-8422

In this paper we characterize the maximal modular ideals of an evolution algebra \(A\,\ \)in order to describe its Jacobson radical, \ \(Rad(A).\) We characterize semisimple evolution algebras (i.e. those such that \(% Rad(A)=\{0\}\))as well as radical ones. We introduce two elemental notions of spectrum of an element \(a\) in an evolution algebra \(A\), namely the spectrum \(% \sigma ^{A}(a)\) and the m-spectrum \(\sigma _{m}^{A}(a)\) (they coincide for associative algebras, but in general \(\sigma ^{A}(a)\subseteq \sigma _{m}^{A}(a),\) and we show examples where the inclusion is strict). We prove that they are non-empty and describe \(\sigma ^{A}(a)\) and \(\sigma _{m}^{A}(a) \) in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of \(A.\) We say \(A\) is m-semisimple (respectively spectrally semisimple) if zero is the unique \ ideal contained into the set of \(a\) in \(A\) such that \(\sigma _{m}^{A}(a)=\{0\}\) \(\ \)(respectively \(\sigma ^{A}(a)=\{0\}\)). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and m-semisimplicity are equivalent)\ we show examples of m-semisimple evolution algebras \(A\) that, nevertheless, are radical algebras (i.e. \(Rad(A)=A\)). Also some theorems about automatic continuity of homomorphisms will be considered.