We review the most important concepts and result regarding the integral closure of ideals and its connection with the multiplicity theory of ideals and modules. We will also see some applications of these notions to the study of singularities of complex analytic maps. In particular, we will recall the celebrated results of Teissier and Gaffney about the equisingularity of analytic deformations of complex maps.

We review the most important concepts and result regarding the integral closure of ideals and its connection with the multiplicity theory of ideals and modules. We will also see some applications of these notions to the study of singularities of complex analytic maps. In particular, we will recall the celebrated results of Teissier and Gaffney about the equisingularity of analytic deformations of complex maps.