The Jacobson radical of an evolution algebra

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 $\sigma^{A}(a)$ 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.