Deep neural networks (DNNs) have made great breakthroughs and seen applications in many domains. However, the incomparable accuracy of DNNs is achieved with the cost of considerable memory consumption and high computational complexity, which restricts their deployment on conventional desktops and portable devices. To address this issue, low-rank factorization, which decomposes the neural network parameters into smaller sized matrices or tensors, has emerged as a promising technique for network compression. In this article, we propose leveraging the emerging tensor ring (TR) factorization to compress the neural network. We investigate the impact of both parameter tensor reshaping and TR decomposition (TRD) on the total number of compressed parameters. To achieve the maximal parameter compression, we propose an algorithm based on prime factorization that simultaneously identifies the optimal tensor reshaping and TRD. In addition, we discover that different execution orders of the core tensors result in varying computational complexities. To identify the optimal execution order, we construct a novel tree structure. Based on this structure, we propose a top-to-bottom splitting algorithm to schedule the execution of core tensors, thereby minimizing computational complexity. We have performed extensive experiments using three kinds of neural networks with three different datasets. The experimental results demonstrate that, compared with the three state-of-the-art algorithms for low-rank factorization, our algorithm can achieve better performance with much lower memory consumption and lower computational complexity.