Question

Prove that every holomorphic image of a ring is isomorphic to some ring quotient​

Answer 1

Step-by-step explanation:

Definition 1. Let R = (R, +R, ·R) and (S, +S, ·S) be rings. A set map φ: R → S is a (ring)

homomorphism if

(1) φ(r1 +R r2) = φ(r1) +S φ(r2) for all r1, r2 ∈ R,

(2) φ(r1 ·R r2) = φ(r1) ·S φ(r2) for all r1, r2 ∈ R, and

(3) φ(1R) = 1S.

For simplicity, we will often write conditions (1) and (2) as φ(r1 + r2) = φ(r1) + φ(r2) and

φ(r1r2) = φ(r1)φ(r2) with the particular addition and multiplication implicit.