A ring homomorphism is a map f:R->S between two rings such that 1. Addition is preserved:f(r_1 + r_2) = f(r_1) + f(r_2), 2. The zero element is mapped to zero: f(0_R) = 0_S, and 3. Multiplication is preserved: f(r_1 r_2) = f(r_1) f(r_2), where the operations on the left-hand side is in R and on the right-hand side in S. Note that a homomorphism must preserve the additive inverse map because f(g) + f(-g) = f(g - g) = f(0_R) = 0_S so -f(g) = f(-g).

group homomorphism | homomorphism | isomorphism | ring | unit ring