|
13-XX
|
Commutative rings and algebras
|
|
13-00
|
General reference works (handbooks, dictionaries, bibliographies, etc.)
|
|
13-01
|
Instructional exposition (textbooks, tutorial papers, etc.)
|
|
13-02
|
Research exposition (monographs, survey articles)
|
|
13-03
|
Historical {!must also be assigned at least one classification number from section 01}
|
|
13-04
|
Explicit machine computation and programs (not the theory of computation or programming)
|
|
13-06
|
Proceedings, conferences, collections, etc.
|
|
13Axx
|
General commutative ring theory
|
|
13A02
|
Graded rings
[See also
16W50]
|
|
13A05
|
Divisibility
|
|
13A10
|
Radical theory
|
|
13A15
|
Ideals; multiplicative ideal theory
|
|
13A18
|
Valuations and their generalizations
[See also
12J20]
|
|
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
[See also
13B22]
|
|
13A50
|
Actions of groups on commutative rings; invariant theory
[See also
14L25]
|
|
13A99
|
None of the above, but in this section
|
|
13Bxx
|
Ring extensions and related topics
|
|
13B02
|
Extension theory
|
|
13B05
|
Galois theory (commutative rings)
|
|
13B10
|
Morphisms
|
|
13B21
|
Integral dependence
|
|
13B22
|
Integral closure of rings and ideals; integrally closed rings, related rings (Japanese, etc.)
|
|
13B24
|
Going up; going down; going between
|
|
13B25
|
Polynomials over commutative rings
[See also
11C08,
13F20,
13M10]
|
|
13B30
|
Quotients and localization
|
|
13B35
|
Completion
[See also
13J10]
|
|
13B40
|
\'Etale and flat extensions; Henselization; Artin approximation
[See also
13J15,
14B12,
14B25]
|
|
13B99
|
None of the above, but in this section
|
|
13Cxx
|
Theory of modules and ideals
|
|
13C05
|
Structure, classification theorems
|
|
13C10
|
Projective and free modules and ideals
[See also
19A13]
|
|
13C11
|
Injective and flat modules and ideals
|
|
13C12
|
Torsion modules and ideals
|
|
13C13
|
Other special types
|
|
13C14
|
Cohen - Macaulay modules
[See also
13H10]
|
|
13C15
|
Dimension theory, depth, related rings (catenary, etc.)
|
|
13C20
|
Class groups
[See also
11R29]
|
|
13C40
|
Linkage, complete intersections and determinantal ideals
[See also
14M12]
|
|
13C99
|
None of the above, but in this section
|
|
13Dxx
|
Homological methods
{For noncommutative rings, see
16Exx;
for general categories, see
18Gxx}
|
|
13D02
|
Syzygies and resolutions
|
|
13D03
|
(Co)homology of commutative rings and algebras (e.g., Hochschild, Andr\'e - Quillen, cyclic, dihedral, etc.)
|
|
13D05
|
Homological dimension
|
|
13D07
|
Homological functors on modules (Tor, Ext, etc.)
|
|
13D10
|
Deformations and infinitesimal methods
[See also
14B10,
14B12,
14D15,
32Gxx]
|
|
13D15
|
Grothendieck groups, K-theory
[See also
14C35,
18F30,
19Axx,
19D50]
|
|
13D22
|
Homological conjectures (intersection theorems)
|
|
13D25
|
Complexes
|
|
13D30
|
Torsion theory
[See also
13C12,
18E40]
|
|
13D40
|
Hilbert - Samuel and Hilbert - Kunz functions; Poincar\'e series
|
|
13D45
|
Local cohomology
[See also
14B15]
|
|
13D99
|
None of the above, but in this section
|
|
13Exx
|
Chain conditions, finiteness conditions
|
|
13E05
|
Noetherian rings and modules
|
|
13E10
|
Artinian rings and modules, finite-dimensional algebras
|
|
13E15
|
Rings and modules of finite generation or presentation; number of generators
|
|
13E99
|
None of the above, but in this section
|
|
13Fxx
|
Arithmetic rings and other special rings
|
|
13F05
|
Dedekind, Pr\"ufer and Krull rings and their generalizations
|
|
13F07
|
Euclidean rings and generalizations
|
|
13F10
|
Principal ideal rings
|
|
13F15
|
Factorial rings, unique factorization domains
[See also
14M05]
|
|
13F20
|
Polynomial rings and ideals; rings of integer-valued polynomials
[See also
11C08,
13B25]
|
|
13F25
|
Formal power series rings
[See also
13J05]
|
|
13F30
|
Valuation rings
[See also
13A18]
|
|
13F40
|
Excellent rings
|
|
13F45
|
Seminormal rings
|
|
13F50
|
Rings with straightening laws, Hodge algebras
|
|
13F55
|
Face and Stanley - Reisner rings; simplicial complexes
[See also
55U10]
|
|
13F99
|
None of the above, but in this section
|
|
13G05
|
Integral domains
|
|
13Hxx
|
Local rings and semilocal rings
|
|
13H05
|
Regular local rings
|
|
13H10
|
Special types (Cohen - Macaulay, Gorenstein, Buchsbaum, etc.)
[See also
14M05]
|
|
13H15
|
Multiplicity theory and related topics
[See also
14C17]
|
|
13H99
|
None of the above, but in this section
|
|
13Jxx
|
Topological rings and modules
[See also
16W60,
16W80]
|
|
13J05
|
Power series rings
[See also
13F25]
|
|
13J07
|
Analytical algebras and rings
[See also
32B05]
|
|
13J10
|
Complete rings, completion
[See also
13B35]
|
|
13J15
|
Henselian rings
[See also
13B40]
|
|
13J20
|
Global topological rings
|
|
13J25
|
Ordered rings
[See also
06F25]
|
|
13J30
|
Real algebra
[See also
12Dxx,
14Pxx]
|
|
13J99
|
None of the above, but in this section
|
|
13K05
|
Witt vectors and related rings
|
|
13L05
|
Applications of logic to commutative algebra
[See also
03Cxx,
03Hxx]
|
|
13Mxx
|
Finite commutative rings
{For number-theoretic aspects, see
11Txx}
|
|
13M05
|
Structure
|
|
13M10
|
Polynomials
|
|
13M99
|
None of the above, but in this section
|
|
13Nxx
|
Differential algebra
[See also
12H05,
14F10]
|
|
13N05
|
Modules of differentials
[See also
16S32]
|
|
13N10
|
Rings of differential operators and their modules
[See also
16S32,
32C38]
|
|
13N15
|
Derivations
|
|
13N99
|
None of the above, but in this section
|
|
13Pxx
|
Computational aspects of commutative algebra
[See also
68W30]
|
|
13P05
|
Polynomials, factorization
[See also
12Y05]
|
|
13P10
|
Polynomial ideals, Gr\"obner bases
[See also
13F20]
|
|
13P99
|
None of the above, but in this section
|