Can a ring have multiple zero elements?

Question

Hi my question is exactly as the title states, I am curious as to this as I am developing a proof which relies heavily on the fact that there can only be one zero element in a ring.Thanks for the help.

Answer 1

If $a$ and $b$ are zero elements. then $a=ab=b$.

Answer 2

If $a,b$ are zeroes, then since $a$ is zero $a+b=b$. Since $b$ is zero $a+b=a$.