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Mathematics Subject Classification – MSC2020

13-XX

Commutative algebra 37713

13-00

General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to commutative algebra 12

13-01

Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 618

13-02

Research exposition (monographs, survey articles) pertaining to commutative algebra 387

13-03

History of commutative algebra [Consider also classification numbers pertaining to Section 01-XX] 149

13-04

Software, source code, etc. for problems pertaining to commutative algebra 270

13-06

Proceedings, conferences, collections, etc. pertaining to commutative algebra 403

13-11

Research data for problems pertaining to commutative algebra 2

13Axx

General commutative ring theory 9662

13A02

Graded rings [See also 16W50] 749

13A05

Divisibility and factorizations in commutative rings [See also 13F15] 702

13A15

Ideals and multiplicative ideal theory in commutative rings 3853

13A18

Valuations and their generalizations for commutative rings [See also 12J20] 867

13A30

Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 1030

13A35

Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure [See also 13B22] 828

13A50

Actions of groups on commutative rings; invariant theory [See also 14L24] 1556

13A70

General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.) [See also 05C25, 05E40] 12

13A99

None of the above, but in this section 662

13Bxx

Commutative ring extensions and related topics 5410

13B02

Extension theory of commutative rings 912

13B05

Galois theory and commutative ring extensions 508

13B10

Morphisms of commutative rings 789

13B21

Integral dependence in commutative rings; going up, going down 209

13B22

Integral closure of commutative rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 846

13B25

Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 1278

13B30

Rings of fractions and localization for commutative rings [See also 16S85] 508

13B35

Completion of commutative rings [See also 13J10] 126

13B40

Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] 148

13B99

None of the above, but in this section 336

13Cxx

Theory of modules and ideals in commutative rings 7614

13C05

Structure, classification theorems for modules and ideals in commutative rings 1671

13C10

Projective and free modules and ideals in commutative rings [See also 19A13] 1393

13C11

Injective and flat modules and ideals in commutative rings 650

13C12

Torsion modules and ideals in commutative rings 250

13C13

Other special types of modules and ideals in commutative rings 941

13C14

Cohen-Macaulay modules [See also 13H10] 936

13C15

Dimension theory, depth, related commutative rings (catenary, etc.) 1382

13C20

Class groups [See also 11R29] 228

13C40

Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 758

13C60

Module categories and commutative rings 96

13C70

Theory of modules and ideals in commutative rings described by combinatorial properties [See also 05C25, 05E40] 0

13C99

None of the above, but in this section 548

13Dxx

Homological methods in commutative ring theory {For noncommutative rings, see 16Exx; for general categories, see 18Gxx} 7905

13D02

Syzygies, resolutions, complexes and commutative rings 1961

13D03

(Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 982

13D05

Homological dimension and commutative rings 965

13D07

Homological functors on modules of commutative rings (Tor, Ext, etc.) 575

13D09

Derived categories and commutative rings 162

13D10

Deformations and infinitesimal methods in commutative ring theory [See also 14B10, 14B12, 14D15, 32Gxx] 317

13D15

Grothendieck groups, \(K\)-theory and commutative rings [See also 14C35, 18F30, 19Axx, 19D50] 621

13D22

Homological conjectures (intersection theorems) in commutative ring theory 102

13D30

Torsion theory for commutative rings [See also 13C12, 18E40] 147

13D40

Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 1465

13D45

Local cohomology and commutative rings [See also 14B15] 1282

13D99

None of the above, but in this section 362

13Exx

Chain conditions, finiteness conditions in commutative ring theory 3491

13E05

Commutative Noetherian rings and modules 1919

13E10

Commutative Artinian rings and modules, finite-dimensional algebras 794

13E15

Commutative rings and modules of finite generation or presentation; number of generators 767

13E99

None of the above, but in this section 355

13Fxx

Arithmetic rings and other special commutative rings 9683

13F05

Dedekind, Prüfer, Krull and Mori rings and their generalizations 1990

13F07

Euclidean rings and generalizations 194

13F10

Principal ideal rings 333

13F15

Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05] 777

13F20

Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 3408

13F25

Formal power series rings [See also 13J05] 1101

13F30

Valuation rings [See also 13A18] 715

13F35

Witt vectors and related rings 183

13F40

Excellent rings 62

13F45

Seminormal rings 102

13F50

Rings with straightening laws, Hodge algebras 99

13F55

Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes [See also 55U10] 818

13F60

Cluster algebras 705

13F65

Commutative rings defined by binomial ideals, toric rings, etc. [See also 14M25] 22

13F70

Other commutative rings defined by combinatorial properties 0

13F99

None of the above, but in this section 334

13Gxx

Integral domains 1832

13G05

Integral domains 1821

13G99

None of the above, but in this section 7

13Hxx

Local rings and semilocal rings 5050

13H05

Regular local rings 868

13H10

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 2816

13H15

Multiplicity theory and related topics [See also 14C17] 1107

13H99

None of the above, but in this section 794

13Jxx

Topological rings and modules [See also 16W60, 16W80] 1997

13J05

Power series rings [See also 13F25] 563

13J07

Analytical algebras and rings [See also 32B05] 46

13J10

Complete rings, completion [See also 13B35] 396

13J15

Henselian rings [See also 13B40] 294

13J20

Global topological rings 107

13J25

Ordered rings [See also 06F25] 145

13J30

Real algebra [See also 12D15, 14Pxx] 268

13J99

None of the above, but in this section 290

13Lxx

Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 303

13L05

Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 301

13L99

None of the above, but in this section 2

13Mxx

Finite commutative rings {For number-theoretic aspects, see 11Txx} 504

13M05

Structure of finite commutative rings 257

13M10

Polynomials and finite commutative rings 142

13M99

None of the above, but in this section 111

13Nxx

Differential algebra [See also 12H05, 14F10] 1665

13N05

Modules of differentials 621

13N10

Commutative rings of differential operators and their modules [See also 16S32, 32C38] 467

13N15

Derivations and commutative rings 522

13N99

None of the above, but in this section 99

13Pxx

Computational aspects and applications of commutative rings [See also 14Qxx, 68W30] {For software etc., see 13-04} 4608

13P05

Polynomials, factorization in commutative rings [See also 12-08] 400

13P10

Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 3294

13P15

Solving polynomial systems; resultants 333

13P20

Computational homological algebra [See also 13Dxx] 70

13P25

Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) 328

13P99

None of the above, but in this section 365

Overview

General and overarching topics; collections

History and biography

Mathematical logic and foundations

Combinatorics

Order, lattices, ordered algebraic structures

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Commutative algebra

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Nonassociative rings and algebras

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Mathematics Subject Classification – MSC2020

Overview