Let R and R' be rings and let N and N' be ideals of R and R', respectively. Let ∅ be a homomorphism or R into R'. Show that ∅ induces a natural homomorphism ∅_* : R/N → R'/N' if ∅[N] ⊆ N'. (Use Exercise 39 of Section 14.)

Data from Exercise 39 of Section 14

Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively. Let ∅ be a homomorphism of G into G'. Show that ∅ induces a natural homomorphism ∅_* : (G/H) → (G'/H') if ∅[H] ⊆ H'. (This fact is used constantly in algebraic topology.)