环与代数：2025–2026 年度综述

An annual survey of the math.RA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Classification of Associative, Poisson, and Leibniz Algebras (17.9%) — Focuses on the classification, structural properties, and representation theory of associative, Poisson, and Leibniz algebras, including cluster structures and quantum affine algebras.
• Finite Structures, Semigroups, and Coding Theory (13.2%) — Investigates finite fields, semigroups, semirings, and their applications in constructing error-correcting codes and analyzing finite-dimensional algebraic structures.
• Matrix Algebras, Polynomials, and Structural Factorizations (11.6%) — Examines matrix algebras, polynomial identities, Jordan structures, and factorization theorems within graded and non-commutative settings.
• Matrix Analysis, Orthogonality, and Linear Systems (10.0%) — Addresses problems related to matrix positivity, orthogonal equivalence, linear system solving, and operator theory in various algebraic spaces.
• Group, Monoid, and Quantum Affine Structures (9.9%) — Explores the structure and cohomology of groups, monoids, and quantum affine spaces, including automorphisms and power monoid arithmetic.
• Hopf Algebras, Modules, and Homological Categories (9.6%) — Studies Hopf algebras, Hopf modules, and their homological properties within abelian and Gorenstein categories, including Galois extensions.
• Lie Algebras, Superalgebras, and Geometric Integrations (8.2%) — Analyzes Lie algebras and superalgebras, focusing on solvable/nilpotent structures, derivations, and applications in geometric numerical integration.
• Commutative, Group, and Polynomial Ring Theory (7.9%) — Investigates commutative rings, group rings, and polynomial rings, with emphasis on grading, separability, and ideal structures.
• Rota-Baxter Operators, Yang-Baxter Equations, and Braces (6.0%) — Focuses on Rota-Baxter and Yang-Baxter operators, skew braces, and their connections to Leibniz algebras and non-associative structures.
• Free Algebras, Hopf Invariants, and Supertropical Structures (5.7%) — Examines free algebras, Hopf algebra invariants, and exceptional structures like supertropical algebras and octonion-related Lie algebras.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.