A note on completeness in the theory of strongly clean rings

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we prove that if $R$ is a ring which is complete with respect to an ideal $I$ and if $x$ is an element of $R$ whose image in $R\mathrm{/}I$ is strongly $\pi$-regular, then $x$ is strongly clean in $R$.