Given $ a * b =a+ab$
Let e be an identity element
We know that $a \ast e = a, a \neq 0$
Now from the given definition of operation * we have a*e =a + ae => a=a+ae
Also e*a = a => a=e+ae
Since $a \ast e$ is not equal to $ e \ast a$ , the operation given by $ a * b =a+ab$ has no identity.