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**►****Foundations****▼****Algebra****►****11-XX Number theory****►****12-XX Field theory and polynomials****►****13-XX Commutative rings and algebras****►****14-XX Algebraic geometry****►****15-XX Linear and multilinear algebra; matrix theory****▼****16-XX Associative rings and algebras****►****16-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****16Bxx General and miscellaneous****►****16Dxx Modules, bimodules and ideals****►****16Exx Homological methods****►****16Gxx Representation theory of rings and algebras**- 16H05 Orders and arithmetic, separable algebras, Azumaya algebras
**►****16Kxx Division rings and semisimple Artin rings****►****16Lxx Local rings and generalizations****►****16Nxx Radicals and radical properties of rings****▼****16Pxx Chain conditions, growth conditions, and other forms of finiteness**- 16P10 Finite rings and finite-dimensional algebras
- 16P20 Artinian rings and modules
- 16P40 Noetherian rings and modules
- 16P50 Localization and Noetherian rings
- 16P60 Chain conditions on annihilators and summands: Goldie-type conditions
- 16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence
- 16P90 Growth rate, Gelfand-Kirillov dimension
- 16P99 None of the above, but in this section

**►****16Rxx Rings with polynomial identity****►****16Sxx Rings and algebras arising under various constructions****►****16Uxx Conditions on elements****►****16Wxx Rings and algebras with additional structure****►****16Yxx Generalizations**- 16Z05 Computational aspects of associative rings

**►****17-XX Nonassociative rings and algebras****▼****18-XX Category theory; homological algebra****►****18-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****▼****18Axx General theory of categories and functors**- 18A05 Definitions, generalizations
- 18A10 Graphs, diagram schemes, precategories
- 18A15 Foundations, relations to logic and deductive systems
- 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
- 18A22 Special properties of functors (faithful, full, etc.)
- 18A23 Natural morphisms, dinatural morphisms
- 18A25 Functor categories, comma categories
- 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
- 18A32 Factorization of morphisms, substructures, quotient structures, congruences, amalgams
- 18A35 Categories admitting limits (complete categories), functors preserving limits, completions
- 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
- 18A99 None of the above, but in this section

**►****18Bxx Special categories****►****18Cxx Categories and theories****▼****18Dxx Categories with structure**- 18D05 Double categories,
*2*-categories, bicategories and generalizations - 18D10 Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
- 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
- 18D20 Enriched categories (over closed or monoidal categories)
- 18D25 Strong functors, strong adjunctions
- 18D30 Fibered categories
- 18D35 Structured objects in a category (group objects, etc.)
- 18D50 Operads
- 18D99 None of the above, but in this section

- 18D05 Double categories,
**►****18Exx Abelian categories****►****18Fxx Categories and geometry****▼****18Gxx Homological algebra**- 18G05 Projectives and injectives
- 18G10 Resolutions; derived functors
- 18G15 Ext and Tor, generalizations, Künneth formula
- 18G20 Homological dimension
- 18G25 Relative homological algebra, projective classes
- 18G30 Simplicial sets, simplicial objects (in a category)
- 18G35 Chain complexes
- 18G40 Spectral sequences, hypercohomology
- 18G50 Nonabelian homological algebra
- 18G55 Homotopical algebra
- 18G60 Other (co)homology theories
- 18G99 None of the above, but in this section

**►****19-XX***K*-theory**▼****20-XX Group theory and generalizations****▼****20-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****20Axx Foundations****►****20Bxx Permutation groups****►****20Cxx Representation theory of groups****►****20Dxx Abstract finite groups****►****20Exx Structure and classification of infinite or finite groups****▼****20Fxx Special aspects of infinite or finite groups**- 20F05 Generators, relations, and presentations
- 20F06 Cancellation theory; application of van Kampen diagrams
- 20F10 Word problems, other decision problems, connections with logic and automata
- 20F12 Commutator calculus
- 20F14 Derived series, central series, and generalizations
- 20F16 Solvable groups, supersolvable groups
- 20F17 Formations of groups, Fitting classes
- 20F18 Nilpotent groups
- 20F19 Generalizations of solvable and nilpotent groups
- 20F22 Other classes of groups defined by subgroup chains
- 20F24 FC-groups and their generalizations
- 20F28 Automorphism groups of groups
- 20F29 Representations of groups as automorphism groups of algebraic systems
- 20F34 Fundamental groups and their automorphisms
- 20F36 Braid groups; Artin groups
- 20F38 Other groups related to topology or analysis
- 20F40 Associated Lie structures
- 20F45 Engel conditions
- 20F50 Periodic groups; locally finite groups
- 20F55 Reflection and Coxeter groups
- 20F60 Ordered groups
- 20F65 Geometric group theory
- 20F67 Hyperbolic groups and nonpositively curved groups
- 20F69 Asymptotic properties of groups
- 20F99 None of the above, but in this section

**►****20Gxx Linear algebraic groups (classical groups)****►****20Hxx Other groups of matrices****►****20Jxx Connections with homological algebra and category theory****►****20Kxx Abelian groups**- 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
**►****20Mxx Semigroups****▼****20Nxx Other generalizations of groups**- 20P05 Probabilistic methods in group theory

**►****Analysis****►****Differential Equations****►****Transformations****►****Geometry****►****Statistics****►****Applied Mathematics****►****Didactics**

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