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Mathematics Subject Classification – MSC2010

13-XX

Commutative algebra 35945

13-00

General reference works (handbooks, dictionaries, bibliographies, etc.) 11

13-01

Instructional exposition (textbooks, tutorial papers, etc.) 602

13-02

Research exposition (monographs, survey articles) 374

13-03

Historical (must also be assigned at least one classification number from Section 01) 143

13-04

Explicit machine computation and programs (not the theory of computation or programming) 238

13-06

Proceedings, conferences, collections, etc. 395

13Axx

General commutative ring theory 9165

13A02

Graded rings [See also 16W50] 690

13A05

Divisibility; factorizations [See also 13F15] 682

13A15

Ideals; multiplicative ideal theory 3646

13A18

Valuations and their generalizations [See also 12J20] 827

13A30

Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 969

13A35

Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure [See also 13B22] 769

13A50

Actions of groups on commutative rings; invariant theory [See also 14L24] 1476

13A99

None of the above, but in this section 646

13Bxx

Ring extensions and related topics 5265

Extension theory 884

Galois theory 501

Morphisms 781

Integral dependence; going up, going down 198

13B22

Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 805

13B25

Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 1245

13B30

Rings of fractions and localization [See also 16S85] 487

13B35

Completion [See also 13J10] 121

13B40

Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] 140

13B99

None of the above, but in this section 324

13Cxx

Theory of modules and ideals 7356

13C05

Structure, classification theorems 1622

13C10

Projective and free modules and ideals [See also 19A13] 1356

13C11

Injective and flat modules and ideals 633

13C12

Torsion modules and ideals 239

13C13

Other special types 893

13C14

Cohen-Macaulay modules [See also 13H10] 884

13C15

Dimension theory, depth, related rings (catenary, etc.) 1349

13C20

Class groups [See also 11R29] 219

13C40

Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 731

13C60

Module categories 82

13C99

None of the above, but in this section 530

13Dxx

Homological methods [For noncommutative rings, see 16Exx; for general categories, see 18Gxx] 7474

13D02

Syzygies, resolutions, complexes 1795

13D03

(Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc. (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 958

13D05

Homological dimension 912

13D07

Homological functors on modules (Tor, Ext, etc.) 518

13D09

Derived categories 143

13D10

Deformations and infinitesimal methods [See also 14B10, 14B12, 14D15, 32Gxx] 307

13D15

Grothendieck groups, \(K\)-theory [See also 14C35, 18F30, 19Axx, 19D50] 611

13D22

Homological conjectures (intersection theorems) 98

13D30

Torsion theory [See also 13C12, 18E40] 138

13D40

Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 1394

13D45

Local cohomology [See also 14B15] 1187

13D99

None of the above, but in this section 359

13Exx

Chain conditions, finiteness conditions 3396

13E05

Noetherian rings and modules 1867

13E10

Artinian rings and modules, finite-dimensional algebras 766

13E15

Rings and modules of finite generation or presentation; number of generators 757

13E99

None of the above, but in this section 344

13Fxx

Arithmetic rings and other special rings 9215

13F05

Dedekind, Prüfer, Krull and Mori rings and their generalizations 1907

13F07

Euclidean rings and generalizations 191

13F10

Principal ideal rings 325

13F15

Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05] 758

13F20

Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 3288

13F25

Formal power series rings [See also 13J05] 1058

13F30

Valuation rings [See also 13A18] 688

13F35

Witt vectors and related rings 168

13F40

Excellent rings 60

13F45

Seminormal rings 97

13F50

Rings with straightening laws, Hodge algebras 96

13F55

Stanley-Reisner face rings; simplicial complexes [See also 55U10] 746

13F60

Cluster algebras 596

13F99

None of the above, but in this section 326

13Gxx

Integral domains 1777

13G05

Integral domains 1770

13G99

None of the above, but in this section 4

13Hxx

Local rings and semilocal rings 4886

13H05

Regular local rings 844

13H10

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 2710

13H15

Multiplicity theory and related topics [See also 14C17] 1063

13H99

None of the above, but in this section 787

13Jxx

Topological rings and modules [See also 16W60, 16W80] 1942

13J05

Power series rings [See also 13F25] 550

13J07

Analytical algebras and rings [See also 32B05] 45

13J10

Complete rings, completion [See also 13B35] 382

13J15

Henselian rings [See also 13B40] 289

13J20

Global topological rings 107

13J25

Ordered rings [See also 06F25] 140

13J30

Real algebra [See also 12D15, 14Pxx] 254

13J99

None of the above, but in this section 286

13Lxx

Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 300

13L05

Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 298

13L99

None of the above, but in this section 2

13Mxx

Finite commutative rings [For number-theoretic aspects, see 11Txx] 477

13M05

Structure 237

13M10

Polynomials 138

13M99

None of the above, but in this section 109

13Nxx

Differential algebra [See also 12H05, 14F10] 1580

13N05

Modules of differentials 609

13N10

Rings of differential operators and their modules [See also 16S32, 32C38] 440

13N15

Derivations 474

13N99

None of the above, but in this section 95

13Pxx

Computational aspects and applications [See also 14Qxx, 68W30] 4296

13P05

Polynomials, factorization [See also 12Y05] 383

13P10

Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 3096

13P15

Solving polynomial systems; resultants 284

13P20

Computational homological algebra [See also 13Dxx] 58

13P25

Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) 274

13P99

None of the above, but in this section 357

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Mathematics Subject Classification – MSC2010

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