  A family $\mathcal{A} \subseteq [\omega]^\omega$ such that for all finite
$\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus
\{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is
said to be ideal independent.
  We prove that an ideal independent family $\mathcal{A}$ is maximal if and
only if $\mathcal A$ is $\mathcal J$-completely separable and maximal $\mathcal
J$-almost disjoint for a particular ideal $\mathcal J$ on $\omega$. We show
that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, where $\mathfrak{s}_{mm}$ is the
minimal cardinality of maximal ideal independent family. This, in particular,
establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given
an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously
adjoin via forcing maximal ideal independent families of cardinality $\lambda$
for each $\lambda\in C$, thus establishing the consistency of $C\subseteq
\hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal
ideal independent family, which remains maximal after forcing with any proper,
$^\omega\omega$-bounding, $p$-point preserving forcing notion and evaluate
$\mathfrak{s}_{mm}$ in several well studied forcing extensions.
  We also study natural filters associated with ideal independence and
introduce an analog of Mr\'owka spaces for ideal independent families.
