# The Jacobson radical of an evolution algebra

### M. Victoria Velasco

Universidad de Granada, Spain

## Abstract

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 $σ_{A}(a)$ and the $m$-spectrum $σ_{m}(a)$ (they coincide for associative algebras, but in general $σ_{A}(a)⊆σ_{m}(a),$ and we show examples where the inclusion is strict). We prove that they are non-empty and describe $σ_{A}(a)$ and $σ_{m}(a)$ 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 $σ_{m}(a)={0}$ (respectively $σ_{A}(a)={0}$). 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.

## Cite this article

M. Victoria Velasco, The Jacobson radical of an evolution algebra. J. Spectr. Theory 9 (2019), no. 2, pp. 601–634

DOI 10.4171/JST/257