Definition:Unital Subalgebra

Definition
Let $R$ be a commutative ring.

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

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

Definition 1
The subalgebra $\left({B_R, \oplus}\right)$ is a unitary subalgebra (or unital subalgebra) of $A_R$ :
 * $1_A \in B$

Definition 2
The subalgebra $\left({B_R, \oplus}\right)$ is a unitary subalgebra (or unital subalgebra) of $A_R$ :
 * $(1) \quad B_R$ has a unit
 * $(2) \quad$That unit is $1_A$.

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

Note
It is possible for $B_R \subseteq A_R$ to be a subalgebra of $A_R$ such that it has a unit $1_B$ such that $1_B \ne 1_A$.

Such a subalgebra is a unitary algebra, but is not a unitary subalgebra of $A_R$.