You are correct that the problem is lack of inverse, however I think your argument is slightly flawed.

0 is not necessarily your identity. I mean, take Z under the operation a.b = a+b-1. Then 1 is your identity.

Here, you have that e=-2x for all , and so e is not fixed (and so no identity exists). Alternatively, you could note that so e=0, and so then you can use the argument you used above. But my point is that you need to prove that e=0, you can't just assume it is.