  A family $\mathscr{I} \subseteq [\omega]^\omega$ such that for all finite
$\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus
\{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is
said to be ideal independent. An ideal independent family which is maximal
under inclusion is said to be a maximal ideal independent family and the least
cardinality of such family is denoted $\mathfrak{s}_{mm}$.
  We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which 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.
