Prime Decompositions of Radicals in Polynomial Rings

Abstract

In this paper we are concerned with the computation of prime decompositions of radicals in polynomial rings over a noetherian commutative ring R with identity. We show that prime decomposition algorithms in R can be lifted to R[x] if for every prime ideal P in R univariate polynomials can be factored over the quotient field of the residue class ring R/P. In the proof of this result a lifting algorithm is constructed which can be considered as a generalization of the algorithm of Ritt and Wu.