We know that a set $F$ together with two binary operations $+$ and $×$ is a field if and only if:

1. addition $+$ is associative;
2. addition $+$ is commutative;
3. addition $+$ has a neutral element $0$;
4. For every $a∈F$ there exists $b∈F$ such that $a+b=0$.
5. $×$ is associative;
6. $×$ is commutative;
7. $×$ distributes over $+$;
8. There is an element $1≠0$ such that $a×1=a$ for all $a∈F$;
9. For each $a∈F$, if $a≠0$ then there exists $x∈F$ such that $a×x=1$.

But I need to find an example of a set or structure that successfully fulfills all the requirements but the number $8$ (meaning there is not an element $1≠0$ such that $a×1=a$ for all $a∈F$). Any thoughts? Thank you in advance.