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**►****Foundations****▼****Algebra****►****11-XX Number theory****▼****12-XX Field theory and polynomials****►****12-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****▼****12Dxx Real and complex fields****►****12Exx General field theory****►****12Fxx Field extensions****►****12Gxx Homological methods (field theory)****►****12Hxx Differential and difference algebra****▼****12Jxx Topological fields****►****12Kxx Generalizations of fields****▼****12Lxx Connections with logic**- 12Y05 Computational aspects of 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****►****16Rxx Rings with polynomial identity****▼****16Sxx Rings and algebras arising under various constructions**- 16S10 Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
- 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
- 16S20 Centralizing and normalizing extensions
- 16S30 Universal enveloping algebras of Lie algebras
- 16S32 Rings of differential operators
- 16S34 Group rings
- 16S35 Twisted and skew group rings, crossed products
- 16S36 Ordinary and skew polynomial rings and semigroup rings
- 16S37 Quadratic and Koszul algebras
- 16S38 Rings arising from non-commutative algebraic geometry
- 16S40 Smash products of general Hopf actions
- 16S50 Endomorphism rings; matrix rings
- 16S60 Rings of functions, subdirect products, sheaves of rings
- 16S70 Extensions of rings by ideals
- 16S80 Deformations of rings
- 16S90 Maximal ring of quotients, torsion theories, radicals on module categories
- 16S99 None of the above, but in this section

**►****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****▼****18Bxx Special categories**- 18B05 Category of sets, characterizations
- 18B10 Category of relations, additive relations
- 18B15 Embedding theorems, universal categories
- 18B20 Categories of machines, automata, operative categories
- 18B25 Topoi
- 18B30 Categories of topological spaces and continuous mappings
- 18B35 Preorders, orders and lattices (viewed as categories)
- 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
- 18B99 None of the above, but in this section

**▼****18Cxx Categories and theories**- 18C05 Equational categories
- 18C10 Theories (e.g. algebraic theories), structure, and semantics
- 18C15 Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples
- 18C20 Algebras and Kleisli categories associated with monads
- 18C30 Sketches and generalizations
- 18C35 Accessible and locally presentable categories
- 18C50 Categorical semantics of formal languages
- 18C99 None of the above, but in this section

**▼****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**►****19-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****19Axx Grothendieck groups and***K*_{0}**►****19Bxx Whitehead groups and***K*_{1}**►****19Cxx Steinberg groups and***K*_{2}**►****19Dxx Higher algebraic***K*-theory**►****19Exx***K*-theory in geometry**►****19Fxx***K*-theory in number theory**►****19Gxx***K*-theory of forms**►****19Jxx Obstructions from topology****►****19Kxx***K*-theory and operator algebras**►****19Lxx Topological***K*-theory- 19M05 Miscellaneous applications of
*K*-theory

**►****20-XX Group theory and generalizations**

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

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