  Let $\mathcal{A}$ be a commutative $AW^*$-algebra, let $S(\mathcal{A})$ be
the *-algebra of all measurable operators affiliated with $\mathcal{A}$, let
$\mathcal{I}$ be an ideal in $\mathcal{A}$, let $s(\mathcal{I})$ be the support
of the ideal $\mathcal{I}$ and let $\mathbb{Y}$ be a solid subspace in
$S(\mathcal{A})$. The necessary and sufficient conditions of existence of
non-zero band preserving derivations from $\mathcal{I}$ to $\mathbb{Y}$ are
given. We show that, in case when $\mathbb{Y}\subset\mathcal{A}$, or
$\mathbb{Y}$ is a quasi-normed solid space, any band preserving derivation from
$\mathcal{I}$ into $\mathbb{Y}$ is always trivial. At the same time, there
exist non-zero band preserving derivations from $\mathcal{I}$ with values in
$S(\mathcal{A})$, if and only if the Boolean algebra of all projections from
the $AW^*$-algebra $s(\mathcal{I})\mathcal{A}$ is not $\sigma$-distributive.
