Reduction of modules

Abstract

The notion of a reduction of an ideal I of a noetherian ring R and the related notions of the integral closure I* of I in R and the integral dependence of an element x of R on I have proved useful in many situations in commutative algebra. The purpose of this paper is to extend these notions to modules. To motivate the definitions which follow, it will be useful to consider these notions for ideals before extending them to modules.