Gröbner Bases and Primary Decomposition of Polynomial Ideals

We present an algorithm to compute the primary decomposition of any ideal in a polynomial ring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic ideal theoretic operations can be performed using Gr6bner bases and we exploit these constructions to inductively reduce the problem to zero dimensional ideals. Here we again exploit the structure of Gr6bner bases to directly compute the primary decomposition using polynomial factorization. We also show how the reduction process can be applied to computing radicals and testing ideals for primality.