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

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

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