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A homomorphism that kills an ideal factors uniquely through the quotient.
Universal property of quotient rings: Let be a ring, let be an 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.
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
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.