Let I be an ideal on ω, the notion of I-AD family was introduced in [3]. Analogous to the well studied ideal I(A) generated by almost disjoint families, we introduce and investigate the ideal I(I-A). It turns out that some properties of I(I-A) depends on the structure of I. Denoting by a(I) the minimum of the cardinalities of infinite I-MAD families, several characterizations for a(I) ≥ ω 1 will be presented. Motivated by the work in [23], we introduce the cardinality s ω,ω (I), and obtain a necessary condition for s ω,ω (I) = s(I). As an application, we show finally that if a(I) ≥ s(I), then BW property coincides with Helly property.