Finite rings with a large number of zero divisors

If $R$ is an associative ring with $n>1$ left-hand zero divisors, then $|R|\leqslant n^2$. We sharpen this estimate for rings that are nonlocal from the left. We describe nonlocal rings with identity, for which an improved estimate can be obtained, and also rings with the condition $|R|=(n-k)(n-l)$, where $k=1,2$ and $l=0,1$.