Commuting and product-zero probability in finite rings

Let cp® be the probability that two random elements of a finite ring R commute and zp® the probability that the product of two random elements in R is zero. We show that if cp®=e, then there exists a Lie-ideal D in the Lie-ring (R,[.,.]) with e-bounded index and with [D,D] of e-bounded order. If zp®=e, then there exists an ideal D in R with e-bounded index and D^2 of e-bounded order. These results are analogous to the well-known theorem of P. Neumann on the commuting probability in finite groups.