Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$  contains a direct sum of $k$ nonzero ideals$\}=\infty. $ How can we construct an ideal $I$ of $R$ and an element $r\in R\setminus I$ such that the ring $(R/I)_{r+I}$ has infinitely many idempotents, where $(R/I)_{r+I}$ is the localization of the ring $R/I$ at the subset $\{r^n+I\mid n\in \mathbb{N}\}$ of $R/I$?