Ring (mathematics)

A ring is a set with addition and multiplication, where addition forms an Abelian group and multiplication is associative, with multiplication distributing over addition.

Abelian group: A group whose operation is commutative. Ring addition must make the elements into an Abelian group.
Ideal (ring theory): An additive subgroup of a ring stable under multiplication by arbitrary ring elements. Ideals are the subsets that make ring quotients well-defined.
Integer ring: The ring of integers under ordinary addition and multiplication. It is the basic example of a commutative ring with identity.
Field (mathematics): A commutative ring with identity in which every nonzero element has a multiplicative inverse. Fields satisfy stronger division properties than general rings.
Monoid: A set with an associative binary operation and an identity element. Ring multiplication is associative and may also have an identity.
Ring homomorphism: A function between rings that preserves addition and multiplication, and usually the multiplicative identity. Homomorphisms compare rings while preserving their operations.
Polynomial ring: A ring whose elements are polynomials with coefficients in a specified ring. Polynomial rings extend ring operations to formal expressions and equations.
Commutative ring: A ring whose multiplication is commutative. This common subclass adds symmetry absent from general ring multiplication.
Distributive property: An algebraic law connecting multiplication and addition by distributing multiplication across sums. Distributivity links the ring's two operations.
Quotient ring: A ring formed by identifying elements whose difference belongs to a fixed ideal. Factoring out an ideal produces a smaller ring that retains ring operations.

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