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**►****Foundations****▼****Algebra****►****11-XX Number theory****►****12-XX 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**- 13B02 Extension theory
- 13B05 Galois theory
- 13B10 Morphisms
- 13B21 Integral dependence
- 13B22 Integral closure of rings and ideals
- 13B24 Going up; going down; going between
- 13B25 Polynomials over commutative rings
- 13B30 Quotients and localization
- 13B35 Completion
- 13B40 Étale and flat extensions; Henselization; Artin approximation
- 13B99 None of the above, but in this section

**►****13Cxx Theory of modules and ideals****▼****13Dxx Homological methods**- 13D02 Syzygies and resolutions
- 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
- 13D05 Homological dimension
- 13D07 Homological functors on modules (Tor, Ext, etc.)
- 13D10 Deformations and infinitesimal methods
- 13D15 Grothendieck groups,
*K*-theory - 13D22 Homological conjectures (intersection theorems)
- 13D25 Complexes
- 13D30 Torsion theory
- 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
- 13D45 Local cohomology
- 13D99 None of the above, but in this section

**►****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****►****18Dxx Categories with structure****►****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****►****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****►****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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