bibkey: costoya2026commutative authors: Cristina Costoya; Amir Fernández Ouaridi; Antonio Viruel year: 2026 title: “Commutative algebras are ideals of evolution algebras” doi: 10.48550/arXiv.2609.32784 url: https://arxiv.org/abs/2609.32784v1 claim: “Finite-dimensional commutative algebras embed in evolution algebras; evolution envelopes and quadratic-map Waring decompositions control embedding dimension.” strata_touched:

• D5/S3/QuadraticForms/EvolutionAlgebras/BareiRemarkThreeFive license: citation-only triage: anchor

Commutative algebras are ideals of evolution algebras

Theorems 2.1 and 3.5 supply embedding constructions. Proposition 3.8 transports perfectness and derived structure. Remark 3.9 starts from a known two-dimensional intrinsic algebra. Section 4 relates multiplication to its quadratic map by polarization. The explicit realization here is in this framework’s spirit; neither the framework nor its published counterexamples is claimed as new.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2609.32784
• URL: https://arxiv.org/abs/2609.32784v1 (v1 submitted 2026-09-26; PDF retrieved 2026-10-04).
• Location: Theorems 2.1 and 3.5, Proposition 3.8, Remark 3.9 and §4 (Waring-rank framework).

Reading and scope

Fresh source retrieval on 2026-10-04 returned HTTP 200. The relevant passages were read for the bounded literature check in #12879. No higher-dimensional minimal idempotent-subspace answer was found in the inspected text. This is not-found-in-searched-scope, not exhaustive absence or a priority claim.