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] /:> [See also 12J20]
13A20	Brauer groups [See also 12Gxx, 16H05]
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics /:> , Analytic spread
13A35	Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22] // ; Tight closure [See also 13B22] ~ [See also 13Mxx]
13A50	Actions of groups on commutative rings; invariant theory [See also 14L25] Invariant theory [See also 14D25]
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 /:< And derivations
13B15	Ramification theory
13B21	Integral dependence
13B22	Integral closure of rings and ideals; integrally closed rings, related rings (Japanese, etc.) /:> Of rings and ideals
13B24	Going up; going down; going between
13B25	Polynomials over commutative rings [See also 11C08, 13F20, 13M10] /:> [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] \'Etale extensions and Henselization; Artin approximation [See also 13J15, 14B12]
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] /:< , 18G05
13C11	Injective and flat modules and ideals /:< [See also 18G05]
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] /:> [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} (Co)homological methods
13D02	Syzygies and resolutions /:> And resolutions
13D03	(Co)homology of commutative rings and algebras (e.g., Hochschild, Andr\'e-Quillen, cyclic, dihedral, etc.) /:> (E.g., Hochschild, Andr\'e-Quillen, cyclic, dihedral, etc.)
13D05	Homological dimension (Co)homological dimension [See also 18G05]
13D07	Homological functors on modules (Tor, Ext, etc.)
13D10	Deformations and infinitesimal methods [See also 14B10, 14B12, 14D15, 32Gxx] /:< 16S80, /:> 14B10,
13D15	Grothendieck groups, $K$-theory [See also 14C35, 18F30, 19Axx, 19D50] // 19Axx, 19D50 ~ 19-XX
13D22	Homological conjectures (intersection theorems)
13D25	Complexes
13D30	Torsion theory [See also 13C12, 18E40]
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincar\'e series /:> And Hilbert-Kunz
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 /:> ; Number of generators
13E99	None of the above, but in this section
13Fxx	Arithmetic rings and other special rings /:< [See also 12-XX]
13F05	Dedekind, Pr\"ufer and Krull rings and their generalizations Dedekind and Pr\"ufer 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] /:> ; Rings of integer-valued polynomials /:> , 13B25
13F25	Formal power series rings [See also 13J05]
13F30	Valuation rings [See also 13A18] /:> [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] /:> [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 /:< (Commutative rings)
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] /:> And their modules
13N15	Derivations
13N99	None of the above, but in this section
13Pxx	Computational aspects of commutative algebra [See also 68W30] // 68W30 ~ 68Q40
13P05	Polynomials, factorization [See also 12Y05]
13P10	Polynomial ideals, Gr\"obner bases [See also 13F20] /:> [See also 13F20]
13P99	None of the above, but in this section