This subtopic studies general commutative algebra, covering rings, ideals, modules, and the fundamental structural ideas used throughout the subject.

This subtopic studies ring extensions and related structures, including integral extensions, normalization, and the behavior of algebraic objects under extension.

This subtopic studies the theory of modules and ideals, analyzing decomposition, homological properties, and the interplay between ring structure and module behavior.

This subtopic studies homological methods in commutative algebra, including projective resolutions, depth, regularity, and the use of derived techniques.

This subtopic studies chain conditions and finite conditions, focusing on Noetherian, Artinian, and related finiteness properties in rings and modules.

This subtopic studies arithmetic and multiplicative structures, including unique factorization, divisibility, and arithmetic properties of commutative rings.

This subtopic studies local rings and related topics, emphasizing localization, maximal ideals, and local-to-global phenomena in commutative algebra.

This subtopic studies topological and ordered rings, where algebra is studied in the presence of additional topological or order structures.

This subtopic studies applications of logic to commutative algebra, connecting algebraic questions with formal systems and definability.

This subtopic studies computational commutative algebra, focusing on algorithms for Gröbner bases, ideals, and algebraic systems arising in practice.