Section IV in a Nutshell

Section IV in a Nutshell This section considers functions from one ring R to another ring S that preserve certain algebraic properties: consider ϕ : R→ S such that

ϕ(a+ b) = ϕ(a) + ϕ(b) and ϕ(ab) = ϕ(a)ϕ(b),

for all a, b ∈ R. We call ϕ a ring homomorphism. A ring homomorphism always preserves the zero of the ring, additive

inverses, unity and multiplicative inverses (Theorem 16.1). While a ring homomorphism ϕ : R→ S need not be onto, it is onto the image of R in S (ϕ(R)) which is itself a subring of S (Theorem 16.2). The kernel of ϕ is defined by

ker(ϕ) = ϕ−1(0) = {r ∈ R : ϕ(r) = 0}.