Let $mathcal{I}$ be an ideal on $omega.$ We define textsf{cov}$^{ast

}left( mathcal{I}right) $ as the least size of a family

$mathcal{Bsubseteq I}$ such that for every infinite $Xinmathcal{I}$ there

is $Binmathcal{B}$ for which $Bcap X$ is infinite. We say an textsf{AD}

family $mathcal{Asubseteq I}$ is a emph{textsf{MAD} family restricted to

}$mathcal{I}$ if for every infinite $Xinmathcal{I}$ there is $Ain

mathcal{A}$such that$leftvert Xcap Arightvert =omega.$ The cardinal

invariant $mathfrak{a}left( mathcal{I}right)$ is defined as the least

size of an infinite textsf{MAD} family restricted to $mathcal{I}.$ The

cardinal invariants $mathfrak{o}$ and $mathfrak{a}_{s}$ may be seen as

particular cases of this class of invariants. In this talk, we will prove that

if the maximum of $mathfrak{a}$ and textsf{cov}$^{ast}left(

mathcal{I}right) $is$omega_{1}$then$mathfrak{a}left( mathcal{I}%

right) =omega_{1}.$ We will obtain some corollaries of this result. This is

part of a joint work with Michael Hruv{s}'{a}k and Osvaldo Tellez.