An operation * has 2 possible identity element:which one's the real identity element?

The rule to find identity element is laid out on Pinter's Abstract Algebra book like this on page 24:


"First solve the equation $\displaystyle x * e = x$ for $\displaystyle e$; if the equation cannot be solved there is no identity element. If it can be solved, it is
still necessary to check that $\displaystyle e * x = x * e = x$ for any $\displaystyle x \in \mathbb{R}$. If it checks, then $\displaystyle e$ is an identity element."


Suppose there is an operation $\displaystyle *$ on $\displaystyle \mathbb{R}$ such that $\displaystyle x * y = \left | x + y \right|$ and we need to find the identity element.

So according to the above rule and by the definition of the operation $\displaystyle *$:

$\displaystyle x * e = \left|x + e \right| = x$

so $\displaystyle x = x + e$ or $\displaystyle x = - (x + e)$

$\displaystyle e = 0$ or $\displaystyle e = 2x$

Now $\displaystyle x * e = x * 0 = \left| x + 0 \right| = \left| x \right| = x$

But $\displaystyle x * e = x * 2x = \left| x + 2x \right | = \left| 3x \right| \neq x $

So can I say that $\displaystyle e = 0$ is an identity element for the operation $\displaystyle *$ because it fulfills the property of identity element?

Can anyone kindly tell me if I'm right or wrong to say that $\displaystyle e = 0$ is an identity element for this operation?

Sorry I got the solution.

You see, if $\displaystyle x = -1$ and $\displaystyle e = 0$ then $\displaystyle x * e = \left | -1 + 0 \right | = 1 \neq -1$

So no $\displaystyle e = 0$ is not an identity element for the operation $\displaystyle *$ on the set $\displaystyle \mathbb{R}$ in this case.

Neither can "2x". An "identity element" for an algebraic stucture, set X with operation *, is member of X such that e+ x= x for any element of X. That is, an identity is a single member of x. There is not a different identity element for each element of X. That operation does not have two identities- it has none.

Thanks HallsofIvy for clarifying the rule for being a identity element.