Epimorphism Preserves Modules/Corollary

Corollary to Epimorphism Preserves Modules

Let $\struct {G, +_G, \circ}_R$ be an unitary $R$-module.

Let $\struct {H, +_H, \circ}_R$ 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 by Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring:

$\forall x \in G: 1_R \circ x = x$

So:

\(\ds 1_R \circ \map \phi x\)

\(=\)

\(\ds \map \phi {1_R \circ x}\)

Definition of $R$-Algebraic Structure Epimorphism

\(\ds \)

\(=\)

\(\ds \map \phi x\)

Definition of Unity of Ring

Thus $H$ is also a unitary module.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations