Quotient ring

Given an ideal $𝔞 ⊴ 𝑅$ , the quotient ring $𝑅 / 𝐼$ can be constructed as the set of cosets of $𝔞$ #m/def/ring

𝑅 / 𝔞 : = {𝑥 + 𝔞 : ∣ : 𝑥 \in 𝑅} (𝑥 𝑦 + 𝔞) = : (𝑥 + 𝔞) (𝑦 + 𝔞)

with canonical projection

𝜋 : 𝑅 ↠ 𝑅 / 𝔞 𝑥 \mapsto 𝑥 + 𝔞 .

Universal property

The quotient ring with the canonical projection is characterized up to unique isomorphism by the universal property:

$𝔞 \subseteq k e r 𝜋$ . If $𝜑 : 𝑅 \to 𝑇$ is a ring homomorphism with $𝔞 \subseteq k e r 𝜑$ , then there exists a unique homomorphism $¯ 𝜑 : 𝑅 / 𝔞 \to 𝑇$ such that $𝜑 = ¯ 𝜑 𝜋$ .

Quotient ring

Universal property

See also