  We construct an example of a unit-regular ring which is not strongly clean,
answering an open question of Nicholson. We also characterize clean matrices
with a zero column, and this allows us to describe an interesting connection
between unit-regular elements and clean elements. It is also proven that given
an element $a$ in a ring $R$, if $a,a^2,\ldots, a^k$ are all regular elements
in $R$ (for some $k\geq 1$), then there exists $w\in R$ such that
$a^{i}w^{i}a^{i}=a^{i}$ for $1\leq i\leq k$, and a similar statement holds for
unit-regular elements. The paper ends with a large number of examples
elucidating further connections (and disconnections) between cleanliness,
regularity, and unit-regularity.
