The results of calculation of the indirect exchange interaction between magnetic layers are presented for the case of a structure with narrow-gap semiconducting IV-VI quantum well. The main mechanism is a magnetic polarization of the size-quantized electrons and holes inside the well. This type of interaction is suggested for the explanation of recent experiments on EuS/PbS structures. The problem of interaction between magnetic layers separated by a non--magnetic material (metal or semiconductor) is becoming important in connection with the recent advances of microtechnology and a search of new device applications. There is a large number of theoretical papers on the coupling between magnetically ordered layers separated by a non-magnetic metal. Several models have been proposed. Despite of different approaches to the problem, the main idea is a transfer of the magnetic polarization from one magnetic layer to another via free carriers of the non-magnetic metal. In other words, the Ruderman-Kittel-KasuyaYoshida (RKKY) mechanism has been always presented in some form. A review of theoretical works has been given in articles [1][2][3] (see also a discussion in [4]).Here we consider a system consisting of a IV-VI based narrow-gap semiconducting quantum well (QW) between two magnetic layers. An example of such system is a EuS/PbS/EuS sandwich structure. The main peculiarity of this structure is that the electron energy spectrum of the material inside the well is of narrow-gap-semiconducting type with interband coupling which includes strong spin-orbit interaction. The quantization of the energy spectrum in a IV-VI based quantum well has been considered in [5].It was shown in a recent experimental work [6] that in superlattices consisting Of EuS/PbS/EuS layers, a ferrOmagnetic phase transition takes place with the *Present address: Center for Quantum Devices, Northwestern University, Evanston IL 60208, USA.(455)