Definition:Unital Subalgebra/Warning

Unital Subalgebra: Warning
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

 * Example of Subalgebra which is Unital but not Unital Subalgebra