The sum-product phenomenon in arbitrary rings

The emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A cdot A$ unless it is in some sense ``close'' to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $R$ and cyclic groups $/q$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $R$. As applications we recover (and generalise) several sum-product theorems already in the literature.