  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)=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(R)=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.
