  We give new a proof of the general Brian\c{c}on-Skoda theorem about ideals of
holomorphic functions by means of multivariable residue calculus. The method
gives new variants of this theorem for products of ideals. Moreover, we obtain
a related result for the ideal generated by the the subdeterminants of a
matrix-valued generically surjective holomorphic function, generalizing the
duality theorem for a complete intersection. We also provide explicit versions
of the various results, including the general Brian\c{c}on-Skoda theorem, with
integral representation formulas.
