$R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Question

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$ \exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad $$ implies $A$ is unital. I know this is true if A is a weak$^*$ dense ideal of $W^*$-algebra. But what about the general case?

This question on MO.

Answer 1

This question is positively answered on MO.

Even more is true. If $A$ have right divisor of all its elements, then $A$ is unital.