Chinese remainder theorem of modules over multivariate polynomial rings

Abstract

The Chinese remainder theorem plays an important role in the areas of number theory and algebra. In the multivariate polynomial rings, the Chinese remainder theorem can be used to solve systems of multivariate polynomial congruence equations. With the help of algebra theory, polynomials that satisfy the congruence equations can be found. In this paper, we investigate the Chinese remainder theorem of modules over multivariate polynomial rings. Using the theory and method of Gröbner bases for modules, we find the polynomial vector that satisfies the Chinese remainder theorem of modules. Furthermore, we obtain the solutions of systems of congruence equations for modules over polynomial rings.