Subring Module is Module/Special Case/Unitary Module

Theorem

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

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

Let $\struct {R, +, \circ}$ be a ring with unity such that $1_R$ is that unity.

Let $1_R \in S$.

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

Proof

From Subring Module is Module: Special Case, we have that $\struct {R, +, \circ_S}_S$ is an $S$-module.

Then by hypothesis $1_R$ is the unity of $\struct {S, +, \circ_S}$.

Thus $\struct {S, +, \circ_S}$ is also a ring with unity.

It follows from Ring with Unity is Module over Itself that $\struct {R, +, \circ_S}_S$ is a unitary module.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.2$