2026/07/16 by Idrees Alshatnawi, Cristina Costoya, Antonio Viruel
#math.RA
We study the Grothendieck ring of finite-dimensional non-degenerate evolution algebras over a field \mathbbK, 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.