Quotient Ring

Let R 𝑅 R be a Ring, and I 𝐼 I be a Two-sided Ideal of R 𝑅 R . The quotient ring R / I 𝑅 𝐼 R/I is the ring formed by quotienting R 𝑅 R by the equivalence relation

x∼y:=x−y∈Isimilar-to𝑥𝑦assign𝑥𝑦𝐼x\sim y:=x-y\in I

Note that there is a canonical map ι R , I : R → R / I : subscript 𝜄 𝑅 𝐼 → 𝑅 𝑅 𝐼 \iota_{R,I}:R\to R/I that sends each element to its equivalence class under ∼ similar-to \sim .

Quotient rings obey the following universal property: A ring homomorphism f : R → S : 𝑓 → 𝑅 𝑆 f:R\to S is factorized by ι R , I : R → R / I : subscript 𝜄 𝑅 𝐼 → 𝑅 𝑅 𝐼 \iota_{R,I}:R\to R/I if and only if I ⊆ ker ⁡ f 𝐼 kernel 𝑓 I\subseteq\ker f ; if this is the case, then the factorization is unique.

If we unfold this, we note that f ⁢ ( I ) ⊆ { 0 S } 𝑓 𝐼 subscript 0 𝑆 f(I)\subseteq\left\{0_{S}\right\} ; EG: f 𝑓 f must make I 𝐼 I must vanish.

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