Can a binary operation have an identity element when it is not associative and commutative? I tried getting the answers in similar questions, everyone says that it's not necessary, but if $e$ is the identity element for any binary operation $*$, which is not associative and commutative, how can
$$a*e=a=e*a$$
when it is not commutative, i.e. $a*b \ne b*a$?
Even if we get a value by solving $a*e=a$. Will we get the same value by solving
$e*a=a$ ?
Please provide an example.

Actually, given any set $S$ and operation $*$ on it (so possibly neither associative nor commutative), we can simply extend this with a new symbol $\color{red}0$ (i.e., $\color{red}0\notin S$) and on the set $S':=S\cup\{\color{red}0\}$ define an operation $\color{red}*$ by
$$x\color{red}*y:=\begin{cases}x&\text{if }y=\color{red}0\\
y&\text{if }x=\color{red}0\\x*y&\text{otherwise} \end{cases}$$
Then $\color{red}*$ is  not associative/commutative if $*$ is not associative/commutative. But $\color{red}0$ is neutral.
