Ideals which generalize $(v^0)$

We consider ideals $d^0(\mathcal{V})$ which are generalizations of the ideal $(v^0)$. We formulate couterparts of Hadamard's theorem. Then, adopting the base tree theorem and applying Kulpa-Szyma\'nski Theorem, we obtain $cov(d^0(\mathcal{V}))\leq add(d^0(\mathcal{V}))^+$.