Rings in which every infinite subset contains a pair of elements with zero product

B. H. Neumann has shown that every infinite subset of a group G contains a pair of commuting elements if and only if G is finite modulo its centre. Here we consider, analogously, the rings in which each infinite subset contains distinct elements x, y with xy = 0 = yx. We show that the rings in question are those which are finite modulo their annihilators provided that they also satisfy the identity x² ≈ 0, which many (and perhaps all) do.