Let $R$ be an arbitrary ring. Now, we assume that we don't know whether $R$ has the multiplicative identity or not.
I know that $R$ has no zero divisors if and only if the cancellation law holds. So, suppose $R$ has no zero divisors. Consider for a nonzero element $a\in R$, $ab=a$ for some $b\in R$. Now, I want to apply for the cancellation law, but, if so, we have $b=1$, where $1$ is the multiplicative identity of $R$.
I think it is false because we don't know whether the ring has the unity.
Thus, I'm wondering when the cancellation law holds.