On the Grothendieck Ring of Finite-Dimensional Non-Degenerate Evolution Algebras

We study the Grothendieck ring of finite-dimensional non-degenerate evolution algebras over a field $\mathbb{K}$, with addition induced by direct sum and multiplication induced by tensor product.

Although its underlying abelian group is freely generated by indecomposable isomorphism classes, the ring structure is much smaller: tensor products create systematic non-cancellation phenomena and many zero-divisors.

The key invariant is the balance of the directed graph associated with a natural basis.

We prove that, for non-degenerate evolution algebras, this balance is independent of the chosen natural basis.

We then show that balance controls cancellation after tensoring: indecomposable factors of balance $1$ cancel, whereas factors of larger balance produce zero-divisor relations under mild hypotheses.

Finally, we analyze the subring generated by cyclic evolution algebras and prove the predicted zero-divisor criterion for all finite direct sums of cyclic algebras.