  Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if
and only if each $R$-algebra which is a direct limit of module finite
$R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is
strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the
converse holds if either the residue class field of $R$ is algebraically closed
or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also
shown that each $R$-algebra which is locally a direct limit of module-finite
algebras, is strongly clean if $R$ is a $\pi$-regular commutative ring.
