Epimorphism Preserves Modules/Corollary

Corollary to Epimorphism preserves Modules
Let $\left({G, +_G, \circ}\right)_R$ be an unitary $R$-module.

Let $\left({H, +_H, \circ}\right)_R$ be an $R$-algebraic structure.

Let $\phi: G \to H$ be an epimorphism.

Then $H$ is a unitary $R$-module.

Proof
Let $G$ be a unitary $R$-module.

From Epimorphism preserves Modules we have that $H$ is an $R$-module.

Then:


 * Module $(4): \quad \forall x \in G: 1_R \circ x = x$

So:

Thus $H$ is also a unitary module.