  Let $I\subset J$ be monomial ideals of a polynomial algebra $S$ over a field.
Then the Stanley depth of $J/I$ is smaller or equal with the Stanley depth of
$\sqrt{J}/\sqrt{I}$. We give also an upper bound for the Stanley depth of the
intersection of two primary monomial ideals $Q$, $Q'$, which is reached if $Q$,
$Q'$ are irreducible, ht$(Q+Q')$ is odd and $\sqrt{Q}$, $\sqrt{Q'}$ have no
common variable.
