Geodesic
Finite rings with a large number of zero divisors
Diskretnaya Matematika, Tome 4 (1992) no. 2, pp. 45-51
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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 $l=0,1$. 
@article{DM_1992_4_2_a4,
     author = {A. N. Alekseichuk and V. P. Elizarov},
     title = {Finite rings with a~large number of zero divisors},
     journal = {Diskretnaya Matematika},
     pages = {45--51},
     year = {1992},
     volume = {4},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1992_4_2_a4/}
}
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A. N. Alekseichuk; V. P. Elizarov. Finite rings with a large number of zero divisors. Diskretnaya Matematika, Tome 4 (1992) no. 2, pp. 45-51. http://geodesic.mathdoc.fr/item/DM_1992_4_2_a4/