  In this short paper, we will show that the space of real valued uniformly
continuous functions defined on a metric space $(X,d)$ is a ring if and only if
every subset $A\subset X$ has one of the following properties:
  $A$ is Bourbaki-bounded, i.e., every uniformly continuous function on $X$ is
bounded on $A$. $A$ contains an infinite uniformly isolated subset, i.e., there
exist $\delta>0$ and an infinite subset $F\subset A$ such that $d(a,x)\geq
\delta$ for every $a\in F, x\in X$.
