Rings, fields, the Chinese remainder theorem and an extension-Part I: theory

The much celebrated Chinese Remainder Theorem has been widely employed in designing fast computationally efficient algorithms in the field of digital signal processing. It has two versions. One is over a ring of integers and the second is over a ring of polynomials with Coefficients defined over a field. In this research work, we extend the Chinese Remainder Theorem to the case of a ring of polynomials with coefficients defined over a finite ring of integers. The entire work is closely related to the already established results on finite fields. This extension is expected to serve as a keystone in the future design of number-theoretic algorithms for performing some of the most computationally intensive tasks. This approach is superior to the number-theoretic-transforms in the sense that the limitations on both the word length and the sequence length are completely removed. In fact, the number-theoretic-transforms may be considered as a very special case of our general approach. Furthermore, the computations required in this work. Which inherits all the merits of the Chinese Remainder Theorem, can be performed in parallel