  We generalize Mattei's result relative to the Brian\c{c}on-Skoda theorem for
foliations to the family of foliations of the second type. We use this
generalization to establish relationships between the Milnor and Tjurina
numbers of foliations of second type, inspired by the results obtained by Liu
for complex hypersurfaces and we determine a lower bound for the global Tjurina
number of an algebraic curve.
