Magnetic molecular nanoclusters are promising components for the design of new magnetic materials and ultrahigh density recording media. The study of these materials is also important due to the possibility of their use in quantum computers, magnetocaloric and magnetooptic devices. The key feature of these materials determining the possibility of their practical use is the value and character of exchange interactions between magnetic ions. But the study of exchange interaction in clusters with more than two paramagnetic centers is nowadays the subject of an intense research and this problem is far from the complete solution. In this paper on the base of Hubbard model we present microscopic model for the calculation of the exchange interactions in a ferrimagnetic ring taking into account hopping term (t) and on site Coulomb repulsion (U). The Heisenberg exchange interaction term and three-spin exchange interaction terms are obtained. The Heisenberg exchange interaction term has the leading order t2/U and three-spin exchange interaction term has the leading order t4/U3. It is shown that the physical reason for the appearance of a three-spin exchange interaction is a ring hopping between four sites. This model is applied to the interpretation of the experimental data obtained by remagnetization of the ferrimagnetic ring Mn6R6. Remagnetization process in Mn6R6 is described on the base of spin Hamiltonian taking into account Heisenberg and three-spin exchange interactions. Exact calculation of the energy and spin structure of this spin Hamiltonian is made. The performed calculations show that in the framework of the Heisenberg model it is impossible to obtain acceptable agreement between theoretical calculations and experimental data. It is shown that acceptable agreement between experimental data and results of theoretical calculations can be obtained only by taking into account three-spin interaction. In this case the value of the three-spin interaction is rather large and the ratio of three-spin exchange constant and Heisenberg exchange constant may be as much as 0.14. According to our microscopic model this value corresponds to the ratio t/U=0.19. The dependence of the values of critical fields on the ring size is also investigated. It is shown that upper and lower critical fields has rather weak dependence on the ring size. This allows an increase of the accuracy of determination of exchange interactions by investigating the remagnetization process of rings with different size. The key feature for quantum computing is the transition from quantum to classical behaviour as ring size is increased. The calculations made show the strong influence of three-spin interaction on this transition. Due to the three-spin interaction this transition became nonmonotonic. Also apparent shift of transition to the classical behaviour to the larger values of the ring size is observed. This increases the working range for quantum computing.