%0 Journal Article
%T The sum-product phenomenon in arbitrary rings
%A Terence Tao
%J Contributions to Discrete Mathematics
%D 2009
%I University of Calgary
%X 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.
%U http://cdm.math.ucalgary.ca/cdm/index.php/cdm/article/view/160