Composition of Ring Homomorphisms is Ring Homomorphism

Theorem
Let:
 * $\struct {R_1, +_1, \odot_1}$
 * $\struct {R_2, +_2, \odot_2}$
 * $\struct {R_3, +_3, \odot_3}$

be rings.

Let:
 * $\phi: \struct {R_1, +_1, \odot_1} \to \struct {R_2, +_2, \odot_2}$
 * $\psi: \struct {R_2, +_2, \odot_2} \to \struct {R_3, +_3, \odot_3}$

be homomorphisms.

Then the composite of $\phi$ and $\psi$ is also a homomorphism.