Let $f_i$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_i$ such that $\sum g_i{f}_i=1$. The effective versions of this result bound the degrees of the $g_i$ in terms of the degrees of the $f_j$. The aim of this paper is to generalize this to the case when the $f_i$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.