Prime Decompositions of Radicals in Polynomial Rings

In the last twenty years several methods for computing primary decompositions of ideals in multivariate polynomial rings over fields (Seidenberg (1974), Lazard (1985), Kredel (1987), Eisenbud et al. (1992)), the integers (Seidenberg, 1978), factorially closed principal ideal domains (Ayoub (1982), Gianni et al. (1988)) and more general rings (Seidenberg, 1984) have been proposed. A related problem is the computation of the irreducible components of an algebraic variety or, equivalently, the computation of the prime ideals in the prime decomposition of a radical. A well-known method for performing this task in a multivariate polynomial ring over a field of characteristic zero is the Ritt-Wu algorithm based on the computation of characteristic sets (Ritt (1950), Wu (1984)). Another method for solving the same problem can be found in (Wang, 1993). Giusti and Heintz deal with irreducible and equidimensional decompositions of varieties given by polynomials in multivariate polynomial rings over infinite perfect fields (Giusti and Heintz, 1990). In Chistov and Grigor’ev (1983) an irreducible decomposition algorithm is presented and analyzed that works in multivariate polynomial rings over fields which are finitely generated over primitive fields. In the present paper we are concerned with the computation of prime decompositions of radicals in polynomial rings over noetherian commutative rings with identity. The prime ideals computed by the Ritt-Wu algorithm are not represented by bases but by so-called irreducible ascending sets. We will not restrict ourselves to either one of these two possible ways of representing prime ideals but will use the following rather general concept. 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 Spec(R), the set of prime ideals in R. We assume that for a given A ∈ S we can algorithmically decide for every f ∈ R whether f ∈ Rep(A). Then A ∈ S is called a representation of the prime ideal Rep(A) and the pair (S, Rep) is called a system of representations in R.