Commutative ring

A ring in which multiplication is commutative: for every pair of elements a and b, ab = ba. Depending on convention, rings may or may not be required to have a multiplicative identity.

Ring (mathematics): An algebraic structure with addition and multiplication, where addition forms an abelian group and multiplication distributes over addition. Commutative rings are rings with the additional requirement that multiplication commute.
Quotient ring: A ring formed by identifying elements whose difference lies in a chosen ideal. Quotienting by an ideal turns its elements into zero and simplifies ring equations.
Algebraic geometry: The study of geometric spaces defined by polynomial equations. Its affine spaces are built from commutative rings through their prime ideals.
Noncommutative ring: A ring whose multiplication is not commutative for at least one pair of elements. It removes the defining constraint and allows order-sensitive multiplication.
Abelian group: A group whose operation is commutative. The addition operation in every ring forms an abelian group.
Localization (commutative algebra): A construction that makes chosen ring elements invertible by adjoining their formal inverses. Localization focuses a ring around selected elements or prime ideals.
Algebraic number theory: The study of arithmetic in number fields and their rings of integers. Rings of integers provide a central setting for factoring and divisibility questions.
Field (mathematics): A commutative ring with identity in which every nonzero element has a multiplicative inverse. Fields add universal nonzero invertibility to the commutative-ring structure.
Ideal (ring theory): An additive subgroup of a ring closed under multiplication by arbitrary ring elements. Ideals let commutative rings be studied through quotient rings and prime structure.
Prime ideal: A proper ideal P such that ab in P implies a in P or b in P. Prime ideals capture divisibility behavior and define the points of a spectrum.

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