Composition of Ring Monomorphisms is Ring Monomorphism

Theorem
Let:
 * $\struct {R_1, +_1, \circ_1}$
 * $\struct {R_2, +_2, \circ_2}$
 * $\struct {R_3, +_3, \circ_3}$

be rings.

Let:
 * $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$
 * $\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$

be (ring) monomorphisms.

Then the composite of $\phi$ and $\psi$ is also a (ring) monomorphism.

Proof
A ring monomorphism is a ring homomorphism which is also an injection.

From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.

From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.