The Jacobson radical of an evolution algebra

In this paper we characterize the maximal modular ideals of an evolution algebra A\,\in order to describe its Jacobson radical, \ $Rad(A)\mathrm{.}$ 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 $% \sigma ^{A}(a)$andthem-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 $A\mathrm{.}$ 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.