Given $a \ast b = a - b$

let e be an identity element of * operation

By definition of identity element a*e=e*a =a

But by definition of * operation a*e =a-e=> a=a-e => e=-1

and e*a=e-a =>a= e-a => e= 2a

we get two different values for e the identity element

This is not possible as the identity is unique. Therefore the operation defined by $a \ast b = a - b$ has no identity element.