Composition of Ring Isomorphisms is Ring Isomorphism

Theorem

Let:

• $\left({R_1, +_1, \circ_1}\right)$
• $\left({R_2, +_2, \circ_2}\right)$
• $\left({R_3, +_3, \circ_3}\right)$

be rings.

Let:

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

be (ring) isomorphisms.

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.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.4$