commutative algebra

English

Noun

commutative algebra (countable and uncountable, plural commutative algebras)

1. (mathematics) The branch of algebra concerned with commutative rings and objects related to them (such as ideals and modules).
   • 1992, R. Keith Dennis, Claudio Pedrini, Michael R. Stein, editors, Algebraic K-theory, Commutative Algebra, and Algebraic Geometry: Proceedings of the Joint Seminar, American Mathematical Society:
   • 1992, Cornelius Greither, Cyclic Galois Extensions of Commutative Rings, Springer, page vii:

     The subject of these notes is a part of commutative algebra, and is also closely related to certain topics in algebraic number theory and algebraic geometry.

   • 2003, Ragnar-Olaf Buchweitz, Morita contexts, idempotents, and Hochschild cohomology — with applications to invariant rings, Luchezar L. Avramov, Marcel Morales, Marc Chardin, Claudia Polini (editors), Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 27,

     It is not until section 7 that we deal with commutative algebra proper, whereas the sections leading up to it should be seen as advocacy that excursions into noncommutative algebra can help to shed light on problems in commutative algebra.

2. (algebra) Any algebra (mathematical structure) in which the multiplication operation is commutative.
   • 1907, James Byrnie Shaw, Synopsis of Linear Associative Algebra, Carnegie Institution of Washington, page 46:

     Theorem. If the complex of the linearly independent numbers of the form $\zeta a-a\zeta$ be deleted from an algebra, the remaining numbers form a commutative algebra.

   • 2010, M. Loaiza, A. Sánchez-Numgaray, “On C*-Algebras of Super Toeplitz Operators with Radial Symbols”, in Roland V. Duduchava, Israel Gohberg, Sergei M. Grudsky, Vladimir Rabinovich, editors, Recent Trends in Toeplitz and Pseudodifferential Operators, Springer, page 175:

     The existence of commutative algebras of Toeplitz operators is not usual, for example there are not non trivial commutative algebras of Toeplitz operators acting on the Hardy space.

   • 2017, Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmuller Groups: Part 2, American Mathematical Society, page 163:

     We devote this chapter to the study of unitary commutative algebras in dg-modules, in simplicial modules and in cosimplicial modules.

   Hyponym: polynomial ring

Synonyms

• (algebra which has a commutative multiplication operation): Abelian algebra

Related terms

• noncommutative algebra (concerned with rings that are not assumed to be commutative)

Translations

branch of mathematics

• Italian: algebra commutativa f
• Portuguese: álgebra comutativa f
• Russian: коммутати́вная а́лгебра f (kommutatívnaja álgebra)

algebra in which multiplication is commutative

• Italian: algebra commutativa f
• Portuguese: álgebra comutativa f
• Russian: коммутати́вная а́лгебра f (kommutatívnaja álgebra)

Further reading

• Associative algebra on Wikipedia.Wikipedia
• Glossary of commutative algebra on Wikipedia.Wikipedia
• List of commutative algebra topics on Wikipedia.Wikipedia
• Commutative algebra on Encyclopedia of Mathematics
• Commutative Algebra on Wolfram MathWorld