Quotient ring

A quotient ring is formed by identifying ring elements whose difference lies in a chosen ideal. Its elements are cosets of that ideal, with addition and multiplication inherited from the original ring.

Ideal: An additive subgroup of a ring that absorbs multiplication by every ring element. The chosen ideal determines which elements become equivalent in the quotient.
Ring (mathematics): An algebraic structure with addition and multiplication, where addition forms an abelian group and multiplication is associative. A quotient ring inherits its operations and ring laws from this structure.
Integers modulo n: The finite ring of residue classes of integers modulo a positive integer n. It is the quotient of the integers by the ideal generated by n.
Modular arithmetic: Arithmetic on residue classes of integers modulo a fixed positive integer. It is arithmetic inside the quotient ring of integers modulo a principal ideal.
Coset: A translate of a subgroup by a group element. Each quotient-ring element is a coset of the defining ideal.
Ring homomorphism: A function between rings that preserves addition, multiplication, and the multiplicative identity when rings are unital. Homomorphisms connect quotient rings to kernels and images.
Finite field: A field with finitely many elements. Quotienting a polynomial ring by an irreducible polynomial can produce one.
Algebraic number theory: The study of number fields, their rings of integers, and related arithmetic structures. Quotient rings help describe prime ideals and residue fields in number rings.
Congruence relation: An equivalence relation compatible with the operations of an algebraic structure. Congruence modulo an ideal expresses exactly which ring elements are identified.
Kernel (algebra): The elements mapped to zero by a homomorphism of algebraic structures. A ring homomorphism's kernel is an ideal that can define its quotient.

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