let's try to find an identity in a more systematic way:

we want a*x = a.

(it should be obvious that a*b = b*a, so we just need to check for a one-sided identity).

now a*x = ax + a + x. set this equal to a, and try to solve for x:

ax + a + x = a

ax + x = 0

(a+1)x = 0. now, can we divide by a+1? yes we can, because a will not be -1.

x = 0/(a+1) = 0.

are you sure it is impossible to find an inverse?

again, we want b such that a*b = 0. we want to solve for b in terms of a, if we can.

ab + a + b = 0

(a+1)b = -a.....i think you can finish. verify your answer.