Composition of Ring Isomorphisms is Ring Isomorphism

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

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

Proof
A ring isomorphism is a ring homomorphism which is also a bijection.

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

From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.