On some structural properties of evolution algebras

We consider the intersection $\mathfrak{M}(A)$ of all maximal ideals of an evolution algebra $A$ and study the structure of the quotient $A/\M(A)$. In a previous work, maximal ideals have been related to hereditary subsets of a graph associated to the given algebra. We investigate the superfluous members both in the family of maximal ideals and also in the set of hereditary subsets of the associated graphs. By using subdirect products we state a structure theorem for arbitrary evolution algebras (arbitrary dimensions and ground field). Specializing in the perfect finite-dimensional case, we obtain a direct sum decomposition instead of a subdirect product and also a uniqueness property. We also study some examples in which Grassmanians appear in a natural way and others that exhibit a richer structure with a nonzero semisimple part that is non-associative.