Commuting and product-zero probability in finite rings

Abstract

Let [Formula: see text] be the probability that two random elements of a finite ring [Formula: see text] commute and [Formula: see text] the probability that the product of two random elements in [Formula: see text] is zero. We show that if [Formula: see text], then there exists a Lie-ideal [Formula: see text] in the Lie-ring [Formula: see text] with [Formula: see text]-bounded index and with [Formula: see text] of [Formula: see text]-bounded order. If [Formula: see text], then there exists an ideal [Formula: see text] in [Formula: see text] with [Formula: see text]-bounded index and [Formula: see text] of [Formula: see text]-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.