Commuting and product-zero probability in finite rings

Let cp(R) be the probability that two random elements of a finite ring R commute and zp(R) the probability that the product of two random elements in R is zero. We show that if cp(R) = epsilon, then there exists a Lie-ideal D in the Lie-ring (R, [\cdot,\cdot]) with epsilon-bounded index and with [D, D] of epsilon-bounded order. If zp(R) = epsilon, then there exists an ideal D in R with e-bounded index and D2 of e-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.