Definition:Unital Subalgebra

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 $A_R$.

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 unital algebra, but is not a unital subalgebra of $A_R$.

Also see

 * Equivalence of Definitions of Unital Subalgebra
 * Definition:Unital Subring