Units and the restriction to ideals

Question

Suppose we are given a unital algebra $A$ and a central idempotent $a\in Z(a)$, i.e. $a^ 2=a$.

Consider the two-sided ideal $B:=A\cdot a$.

We have that $a$ is the unit in $B$. But the unit in $A$ (which was considered to be unital) is also a unit in $B\subset A$ and a unit is unique. So the unit $a$ is the unit in $A$?

Answer 1

No, suppose that $A$ is the ring of $2\times 2$ diagonal matrices and take $a=\pmatrix{1&0\cr 0&0}$.

Answer 2

The answer of Tsemo Aristide gives a counterexample. And you are wrong because, if $I$ is the unit element of $A$ than $Ia=a\ne I$, so $I$ is not an element of $B$.