This study is concerned with the issue of solving the discrete periodic Riccati matrix equations (DPAREs) in discrete-time periodic linear systems. Currently, many existing results for solving the DPARE involve much matrix inversion operations. In order to diminish the matrix inversion operations, a novel reduced inversion zeroing neural network (RIZNN) model is established by constructing a special group of matrix-value error equations. Besides, a nonlinear activation function (NAF) that combines a hyperbolic sine function with an exponential function is designed to accelerate the convergence rate of the RIZNN model. Specially, with the help of a time-varying function, a prescribed-time convergent RIZNN (PT-RIZNN) model is constructed based on the RIZNN model. The distinctive feature of the PT-RIZNN model is that the setting time can be prescribed a prior. Moreover, the convergence property of the PT-RIZNN model and the superiority of the NAF are theoretically proven, respectively. Simulation results are supplied to demonstrate the effectiveness of the PT-RIZNN model and the superiority of the NAF.