An Algorithm to Compute a Primary Decomposition of Modules in Polynomial Rings over the Integers

TL;DR

This paper introduces an algorithm for primary decomposition of modules over polynomial rings with integer coefficients, extending ideal decomposition techniques to modules and implementing the method in Singular.

Contribution

It develops a novel algorithm for primary decomposition of modules over polynomial rings over integers, utilizing ideal decomposition and pseudo-primary techniques.

Findings

Algorithm successfully computes primary decompositions of modules.

Implementation in Singular demonstrates practical applicability.

Method extends ideal decomposition techniques to modules over integer polynomial rings.

Abstract

We present an algorithm to compute the primary decomposition of a submodule $\mathcal{N}$ of the free module $\Z[x_1, \ldots, x_n]^m$. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of $\mathcal{N}$, i.e. the minimal associated primes of the ideal $\Ann(\Z[x_1, \ldots, x_n]^m /\mathcal{N})$ in $\Z[x_1,\ldots,x_n]$ and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in {\sc Singular}.