Integrally closed modules and their divisors

There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another 'determinantal' ideal obtained through Noether normalization. They are useful to describe the integral closure of some class of modules and to study the completeness of the modules of Kahler differentials.