Definition:Unital Subalgebra/Definition 2

Definition
Let $R$ be a commutative ring.

Let $\left({A_R, *}\right)$ be an unital algebra over $R$ whose unit is $1_A$.

Let $\left({B_R, *}\right)$ be a subalgebra of $A_R$.

The subalgebra $\left({B_R, *}\right)$ is a unital subalgebra of $A_R$ :
 * $(1) \quad B_R$ is unital
 * $(2) \quad$Its unit is $1_A$.

That is, a unital subalgebra of $A_R$ must not only have a unit, but that unit must also be the same unit as that of $A_R$.

Also see

 * Equivalence of Definitions of Unital Subalgebra