Subring Module/Special Case

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Theorem

Let $S$ be a subring of the ring $\struct {R, +, \circ}$.

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


Then $\struct {R, +, \circ_S}_S$ is an $S$-module.


If $\struct {R, +, \circ}$ has a unity, $1_R$, and $1_R \in S$, then $\struct {R, +, \circ_S}_S$ is a unitary $S$-module.


Proof

From Ring is Module over Itself, it follows that:

$\struct {R, +, \circ}_R$ is an $R$-module.
If $\struct {R, +, \circ}$ has a unity, then $\struct {R, +, \circ}_R$ is unitary.


Now the theorem follows directly from Subring Module.

$\blacksquare$


Sources