  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 $\Z/q\Z$. 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.

Combinatorics