We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degenerate evolution algebra. When the algebra is degenerate, the uniqueness cannot be assured.The graph associated to an evolution algebra (relative to a natural basis) will play a fundamental role to describe the structure of the algebra. Concretely, a non-degenerate evolution algebra is irreducible if and only if the graph is connected. Moreover, when the evolution algebra is finite-dimensional, we give a process (called the fragmentation process) to decompose the algebra into irreducible components.2010 Mathematics Subject Classification. Primary 17A60, 05C25.

We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph we prove that the space of derivations is zero. For the remaining families of evolution algebras we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph's structure impacts in the characterization of derivations for a given evolution algebra. Therefore our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible 3-dimensional evolution algebras.

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In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. Central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis.We also consider ideals in perfect evolution algebras and prove that they coincide with the basic ideals.Nilpotent elements of order three can be localized (in a perfect evolution algebra over a field in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary field has a nilpotent element of order three, then its structure matrix has a vanishing principal minor.We finish by considering the adjoint evolution algebra and relating its properties to the corresponding in the initial evolution algebra.