Composition of Ring Monomorphisms is Ring Monomorphism

Theorem
Let: be rings.
 * $\left({R_1, +_1, \circ_1}\right)$
 * $\left({R_2, +_2, \circ_2}\right)$
 * $\left({R_3, +_3, \circ_3}\right)$

Let: be (ring) monomorphisms.
 * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$
 * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$

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.