Prime decompositions of radicals in polynomial rings
Let \(R\) be a noetherian commutative ring with identity, \(S\) a set of finite subsets of \(R\), and Rep a surjective function from \(S\) to \(\text{Spec} (R)\). The author proves the following result: Theorem. Assume that for every \(C\in S\) there exists an algorithm for expressing every nonconstant element of \(K[x]\) as a product of irreducible polynomials, where \(K\) is the quotient field of the residue class ring \(R/ \text{Rep} (C)\). Then there exists a system of representations \((\overline S, \text{Rep})\) in \(R[x]\) and an algorithm that computes for a given finite subset \(F\) of \(R[x]\) a subset \(\{C_1, \ldots, C_r\}\) of \(\overline S\) such that \(\text{Radical} (F) = \bigcap^r_{i = 1} \text{Rep} (C_i)\). The algorithm that is constructed may be considered to be a generalization of the primary decomposition lifting the algorithm of Witt and Wu for multivariate polynomial rings over a field of characteristic zero. The author suggests modifications to his algorithm which would address problems concerning superfluous components and termination times.
scientific article; zbMATH DE number 762066