Composition of Ring Homomorphisms is Ring Homomorphism/Proof 1

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.

Proof

A specific instance of Composite of Homomorphisms on Algebraic Structure is Homomorphism.

$\blacksquare$