There is an operation on a set $\displaystyle \mathbb{R}$ (real numbers) such that:

$\displaystyle x * y = \sqrt{x^2 + y^2}$

Does $\displaystyle \mathbb{R}$ have an identity element with respect to operation $\displaystyle *$?

Now I find the identity element like this:

$\displaystyle \begin{align*}x * e =& \sqrt{x^2 + e^2} \\ x =& \sqrt{x^2 + e^2}......................\text{[by definition of identity element]} \\ x^2 =& (x^2 + e^2) \\ e^2 =& 0 \\ e =& 0 \end{align*}$

So now $\displaystyle x * e = x * 0 = \pm x$

Is this $\displaystyle e = 0$ a valid identity element for this operation?

Because an identity element has a property that it doesn't change the original element when used with the operation $\displaystyle *$ on set $\displaystyle \mathbb{R}$.

$\displaystyle x * e = \sqrt{x^2 + 0^2}$ can be $\displaystyle -x$ The original element is modified.

But it's also possible that $\displaystyle x * e = +x$. The original element is left untouched. I'm confused by these two situations.

So my question is:

Is this $\displaystyle e = 0$ a valid identity element of $\displaystyle \mathbb{R}$ with respect to $\displaystyle *$?