Find a concept
Start typing to search the mathematical index.
A homomorphism that kills an ideal factors uniquely through the quotient.
Universal property of quotient rings: Let be a ring, let be a two-sided ideal, and let be the canonical projection onto the quotient ring. For any ring homomorphism such that (where is the kernel), there exists a unique ring homomorphism with
This property characterizes up to unique isomorphism and is the categorical mechanism behind “imposing relations” by quotienting.
Let be a ring and an additive subgroup. Then is a left ideal if for every , a right ideal if for every , and a two-sided ideal if both conditions hold.
Let be a ring and let be a two-sided ideal of . The quotient ring is the quotient set of by the equivalence relation iff , with addition and multiplication on cosets defined by
The two-sided condition ensures multiplication is well-defined.
The canonical projection is a surjective ring homomorphism with kernel , and satisfies the universal property of quotients.