I'm in the process of doing some practice problems to better understand binary operations. I'm currently working on the following binary operation
On the set, $\mathbb{Z}$, the binary operation $a * b = a + 2b$
I've found that the operation is not associative and commutative (I could be wrong) but now when determining if this operation has an identity element, is it enough that the operation is non commutative and non associative to say that it has no identity element?
I guess to further strengthen the claim, I would look for cases where the identity element fails?
All help is greatly appreciated!
Thanks for looking!
No, you can't say that the operation has no identity just because it is not associative or commutative. To say that $e$ is an identity for the operation is to say that $e*a=a=a*e$ for all $a$. To prove that there is no identity, you would need to prove that no such $e$ exists; there's no reason that an identity automatically can't exist just because the operation is not commutative or associative.
To figure out whether there is an identity, then, you should look at what the equation $e*a=a=a*e$ says. For your operation, it says that $e$ must satisfy $$e+2a=a=a+2e$$ for all $a\in\mathbb{Z}$. Is there any such integer $e$?