Ring theory

Ring theory studies rings: sets with addition and multiplication in which addition forms an abelian group, multiplication is associative, and multiplication distributes over addition.

Group: A set with a binary operation satisfying closure, associativity, an identity element, and inverses. Ring addition is required to form an abelian group.
Ideal: An additive subgroup of a ring that absorbs multiplication by every ring element. Ideals identify subsets that can be collapsed while preserving ring operations.
Algebraic number theory: The study of number fields, their integers, and arithmetic properties using algebraic methods. Rings of integers encode arithmetic beyond the ordinary integers.
Commutative ring: A ring whose multiplication is commutative. Commutativity separates the setting of much classical theory from broader ring theory.
Abelian group: A group whose operation is commutative. The additive structure of every ring is an abelian group.
Quotient ring: A ring formed by identifying elements whose difference belongs to a chosen ideal. Quotient rings turn ideals into kernels of natural projection maps.
Commutative algebra: The study of commutative rings and their modules, ideals, and related structures. It develops ring theory’s tools for commutative rings and their geometry.
Noncommutative ring: A ring whose multiplication is not commutative. Matrix rings show why ring theory cannot assume factors commute.
Monoid: A set with an associative binary operation and an identity element. A ring with identity has a multiplicative monoid.
Ring homomorphism: A map between rings that preserves addition and multiplication. Its kernel is an ideal, linking maps to quotient constructions.

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