Subring Module is Module/Unitary

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Theorem

Let $\struct {R, +, \times}$ be a ring.

Let $\struct {S, +_S, \times_S}$ be a subring of $R$.

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

Let $\circ_S$ be the restriction of $\circ$ to $S \times G$.


Let $\struct {R, +, \times}$ be a ring with unity.

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

Let $1_R \in S$.


Then $\struct{G, +_G, \circ_S}_S$ is also unitary.


Proof

From Subring Module is Module, we have that $\struct {G, +_G, \circ_S}_S$ is an $S$-module.

It remains to be demonstrated that $\struct{G, +_G, \circ_S}_S$ is unitary.


To show this, we must prove that:

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


Since $1_R \in S$ by assumption, the product $1_R \circ_S x$ is defined.

We now have:

\(\ds 1_R \circ_S x\) \(=\) \(\ds 1_R \circ x\)
\(\ds \) \(=\) \(\ds x\) Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring on $\struct {G, +_G, \circ}_R$

and the proof is complete.

$\blacksquare$


Sources

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