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: \begin{itemize} ıem $A$ is Bourbaki-bounded, i.e., every uniformly continuous function on $X$ is bounded on $A$. ıem $A$ 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 $a\in F$, $x\in X\backslash\{a\}$. \end{itemize}