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**►****Foundations****▼****Algebra****▼****11-XX Number theory****►****11-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****11Axx Elementary number theory****►****11Bxx Sequences and sets****►****11Cxx Polynomials and matrices****►****11Dxx Diophantine equations****►****11Exx Forms and linear algebraic groups****►****11Fxx Discontinuous groups and automorphic forms****►****11Gxx Arithmetic algebraic geometry (Diophantine geometry)****►****11Hxx Geometry of numbers****►****11Jxx Diophantine approximation, transcendental number theory****►****11Kxx Probabilistic theory: distribution modulo***1*; metric theory of algorithms**►****11Lxx Exponential sums and character sums****►****11Mxx Zeta and***L*-functions: analytic theory**►****11Nxx Multiplicative number theory****►****11Pxx Additive number theory; partitions****►****11Rxx Algebraic number theory: global fields****►****11Sxx Algebraic number theory: local and***p*-adic fields**►****11Txx Finite fields and commutative rings (number-theoretic aspects)****►****11Uxx Connections with logic****►****11Yxx Computational number theory**- 11Zxx Miscellaneous applications of 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****►****13-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****▼****13Axx General commutative ring theory**- 13A02 Graded rings
- 13A05 Divisibility
- 13A10 Radical theory
- 13A15 Ideals; multiplicative ideal theory
- 13A18 Valuations and their generalizations
- 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
- 13A35 Characteristic
*p*methods (Frobenius endomorphism) and reduction to characteristic*p*; tight closure - 13A50 Actions of groups on commutative rings; invariant theory
- 13A99 None of the above, but in this section

**►****13Bxx Ring extensions and related topics****►****13Cxx Theory of modules and ideals****►****13Dxx Homological methods****►****13Exx Chain conditions, finiteness conditions****►****13Fxx Arithmetic rings and other special rings**- 13G05 Integral domains
**►****13Hxx Local rings and semilocal rings****►****13Jxx Topological rings and modules**- 13K05 Witt vectors and related rings
- 13L05 Applications of logic to commutative algebra
**►****13Mxx Finite commutative rings****►****13Nxx Differential algebra****▼****13Pxx Computational aspects of commutative algebra**

**►****14-XX Algebraic geometry****►****15-XX Linear and multilinear algebra; matrix theory****►****16-XX Associative rings and algebras****►****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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