  We introduce and study a notion of module nuclear dimension for a
$C^{*}$-algebra $A$ which is $C^*$-module over another $C^*$-algebra $\mathfrak
A$ with compatible actions. We show that the module nuclear dimension of $A$ is
zero if $A$ is $\mathfrak A$-NF. The converse is shown to hold when $\mathfrak
A$ is a $C(X)$-algebra with simple fibers, with $X$ compact and totally
disconnected. We also introduce a notion of module decomposition rank, and show
that when $\mathfrak A$ is unital and simple, if the module decomposition rank
of $A$ is finite then $A$ is $\mathfrak A$-QD. We study the set $\mathcal
T_\mathfrak A(A)$ of $\mathfrak A$-valued module traces on $A$ and relate the
Cuntz semigroup of $A$ with lower semicontinuous affine functions on the set
$\mathcal T_\mathfrak A(A)$. Along the way, we also prove a module Choi-Effros
lifting theorem. We give estimates of the module nuclear dimension for a class
of examples.
