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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****►****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**- 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**

**▼****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- 19D06
*Q*- and plus-constructions - 19D10 Algebraic
*K*-theory of spaces - 19D23 Symmetric monoidal categories
- 19D25 Karoubi-Villamayor-Gersten
*K*-theory - 19D35 Negative
*K*-theory, NK and Nil - 19D45 Higher symbols, Milnor
*K*-theory - 19D50 Computations of higher
*K*-theory of rings - 19D55
*K*-theory and homology; cyclic homology and cohomology - 19D99 None of the above, but in this section

- 19D06
**►****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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