In the three-dimensional Hubbard model with a strong Coulomb interaction (U>>t) ((t-J) model), the influence of spin fluctuations on the formation of a superconducting state is investigated. The system's magnetic susceptibility is calculated in terms of the generalized random-phase approximation, using the diagram technique for Hubbard operators. The authors find the magnetic phase transition lines on the (t/U, n) plane, where n is the electron concentration. Equations similar to those in the strong-coupling theory for a superconductor in the (t-J) model are derived taking into account the spin fluctuations. Equations are obtained for the superconducting transition temperatures T_c for the order parameter of s and d symmetries. Spin fluctuations are shown to suppress the superconductivity near the ferromagnetic instability of the paramagnetic phase for both s and d states. Near the antiferromagnetic instability, a sufficient enhancement of T_c is possible.