12L10 Ultraproducts [See also <a href="#code:03C20">03C20</a>]

12L12 Model theory [See also <a href="#code:03C60">03C60</a>]

12L15 Nonstandard arithmetic [See also <a href="#code:03H15">03H15</a>]

12L99 None of the above, but in this section

12Yxx Computational aspects of field theory and polynomials

12Y05 Computational aspects of field theory and polynomials

12Y99 None of the above, but in this section

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.

13A02 Graded rings [See also <a href="#code:16W50">16W50</a>]

13A05 Divisibility; factorizations [See also <a href="#code:13F15">13F15</a>]

13A15 Ideals; multiplicative ideal theory

13A18 Valuations and their generalizations [See also <a href="#code:12J20">12J20</a>]

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 <a href="#code:13B22">13B22</a>]

13A50 Actions of groups on commutative rings; invariant theory [See also <a href="#code:14L24">14L24</a>]

13A99 None of the above, but in this section

13Bxx Ring extensions and related topics

13B21 Integral dependence; going up, going down

13B22 Integral closure of rings and ideals [See also <a href="#code:13A35">13A35</a>]; integrally closed rings, related rings (Japanese, etc.)

13B25 Polynomials over commutative rings [See also <a href="#code:11C08">11C08</a>, <a href="#code:11T06">11T06</a>, <a href="#code:13F20">13F20</a>, <a href="#code:13M10">13M10</a>]

13B30 Rings of fractions and localization [See also <a href="#code:16S85">16S85</a>]

13B35 Completion [See also <a href="#code:13J10">13J10</a>]