Multiplicative identity being equal to additive identity in a field

When $0$ is the additive identity for a ring, for any element $x$ in the ring, $x*0=x*(0+0)=x*0+x*0$. Subtracting $x*0$ from both sides of $x*0=x*0+x*0$ tells you that $x*0=0$.

If now $x$ is posited to be the multiplicative identity, it says that $x=x*0=0$. So such a ring is necessarily $\{0\}$.

As people have mentioned in the comments, the zero ring is excluded as a field by standard field axioms.