Noncommutative algebra

Noncommutative algebra studies algebraic structures in which multiplication need not satisfy the commutative law. Its central objects include rings, algebras, and their modules.

Ring: A set with addition and multiplication satisfying specified algebraic laws, including distributivity. Noncommutative rings are the basic setting for much of the subject.
Matrix algebra: An algebra whose elements are matrices with entries from a field or ring. Matrix multiplication gives a familiar, concrete source of noncommutativity.
Commutative algebra: The study of rings and algebras whose multiplication satisfies the commutative law. Its familiar symmetry is precisely what noncommutative algebra relaxes.
Noncommutative geometry: An approach to geometry that replaces spaces with noncommutative algebras of functions. It treats noncommutative algebras as coordinate systems for generalized spaces.
Associative algebra: A vector space or module equipped with an associative bilinear multiplication. These algebras combine ring multiplication with a scalar field or ring.
Representation theory: The study of algebraic structures through their actions as linear transformations on vector spaces. Representations translate abstract multiplication into operators that can be analyzed.
Polynomial ring: A ring of finite formal sums of variables with coefficients and commutative indeterminates in the usual setting. Ordinary polynomial multiplication permits reorderings that may fail in noncommutative rings.
C*-algebra: A complex Banach algebra with an involution satisfying the C-star identity. C*-algebras provide analytic frameworks for many operator and quantum systems.
Module theory: The study of modules, which generalize vector spaces by allowing scalars from a ring. Modules reveal how noncommutative rings act on structured spaces.
Jacobson radical: The intersection of a ring's maximal left ideals, capturing part of its non-semisimple structure. It measures how far a noncommutative ring is from having well-behaved simple modules.

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